Optimal Control with
MCTDH and CRAB
Masterarbeit
Christian ZiemannUniversität HamburgFachbereich Physik
Erstgutachter: Prof. Dr. R. J. Dwayne Miller (MPI für Struktur und Dynamik der Materie)Zweitgutachter: Prof. Dr. Peter Schmelcher (Institut für Laserphysik)
Abstract
The multi-configurational time-dependent Hartree method (MCTDH) has been established as apowerful tool for the simulation of multi-dimensional systems, especially in the realm of molec-ular physics. It has also been used for optimal control of molecular systems, mostly employingthe well-known Rabitz and the Krotov optimization schemes. In this thesis, the MCTDH soft-ware is combined with the relatively new optimization method CRAB (chopped random basis).State selective excitation of different one-dimensional systems is performed with optimized laserpulses. The main objective was vibrational excitation of a Morse oscillator with a light field ex-panded in a finite Fourier basis for CRAB optimization. In this case CRAB performed at leastas good as the established Krotov scheme, with target populations near 100% despite smallercomputational effort.
Zusammenfassung
Das MCTDH-Verfahren hat sich als mächtiges Werkzeug zur Simulation mehrdimensionalerSysteme bewährt, insbesondere im Bereich der Molekülphysik. Es wurde auch bereits für ge-zielte Kontrolle molekularer Systeme eingesetzt, hauptsächlich unter Verwendung der bekann-ten Optimierungsschemata nach Krotov und Rabitz. In dieser Arbeit wird MCTDH mit dem ver-gleichsweise neuen Optimierungsverfahren CRAB kombiniert. Eigenzustände in verschiedeneneindimensionalen Systeme werden selektiv mit optimierten Laserpulsen angeregt. Hauptaugen-merk liegt auf einem Morseoszillator, dessen Vibrationszustände gezielt angeregt werden. Dasdafür genutzte Feld wird in einer endlichen Fourierbasis dargestellt und mit Hilfe von CRABoptimiert. CRAB erweist sich für diese Anwendung als mindestens gleichwertig zu dem eta-blierten Krotov-Schema, indem die Zielzustände trotz geringem Rechenaufwand zu fast 100%besetzt werden.
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Contents
1 Introduction 7
2 Physics of Molecules 9
2.1 Born-Oppenheimer approximation . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Eigenstates of diatomic molecules . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Molecules in electromagnetic fields . . . . . . . . . . . . . . . . . . . . . . . 172.4 Quantum Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Non-adiabatic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Numerical Methods 24
3.1 MCTDH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.1 Full-CI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 TDH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.3 MCTDH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 CRAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 Nelder-Mead and Subplex methods . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Calculations 37
4.1 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.1.1 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.1.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1.3 Coupling to a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Morse Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.1 Choice of DVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.2.3 State selective excitation . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.1 One- and two-parameter control . . . . . . . . . . . . . . . . . . . . . 484.3.2 CRAB with ten parameters . . . . . . . . . . . . . . . . . . . . . . . . 494.3.3 Frequency optimization . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.4 LiCs diatomic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3.5 Krotov scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
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5 Results and Discussion 53
5.1 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.1 Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.1.2 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.1.3 Coupling to a field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Morse Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.1 Choice of DVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2.3 State selective excitation . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.3.1 One- and two-parameter control . . . . . . . . . . . . . . . . . . . . . 715.3.2 CRAB with ten parameters . . . . . . . . . . . . . . . . . . . . . . . . 735.3.3 Frequency optimization . . . . . . . . . . . . . . . . . . . . . . . . . 795.3.4 LiCs diatomic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795.3.5 Krotov scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Memory and CPU consumption . . . . . . . . . . . . . . . . . . . . . . . . . 82
6 Conclusion 85
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1 Introduction
The past two decades have seen impressive progress in the field of quantum control theory. Itsmain goal may be summarized as the design of optimal external fields to guide a quantum systeminto a desired state. In the field of quantum chemistry, this usually means finding specificallyshaped light pulses to steer a molecular system into a target state. The ultimate goal of thiswould be laser control of chemical reactions. This can be done by experimental methods, butalso theoretical simulations have shown to be useful to understand the underlying dynamics [4].
These theoretical calculations require two main ingredients: The response of the system inquestion to an external field must be modelled accurately (i.e. its wavefunction must be propa-gated in time), which can then be used by an iterative algorithm to optimize said field.
Molecular systems are inherently difficult to simulate numerically due to the high number ofdegrees of freedom (DOFs). Even after employing simplifications such as the Born-Oppenheimerapproximation, the scaling behaviour of multi-dimensional systems discourages exact calcula-tions. A number of numerical methods exist, though, each offering a trade off between accuracyand numerical effort. In this work the Multi Configurational Time Dependent Hartree method(MCTDH) [19] will be used, which has been shown to simulate quantum systems of high di-mensionality very efficiently and with good accuracy[3].
With such a tool for efficient wavefunction propagation at hand, it has to be embedded intoan optimal control algorithm such as the well-probed Krotov scheme, which has already beenimplemented with MCTDH. This thesis will pursue a different scheme, namely the ChoppedRandom Basis (CRAB) method. CRAB [5] has only recently been proposed as a versatile toolfor a wide range of optimal control problems.
Hence this work will attempt to perform optimal control with the CRAB algorithm, usingMCTDH to calculate the system’s behaviour in the external field. The systems under study area one-dimensional Morse oscillator as a simple model for a generic diatomic molecule and theLiCs diatomic. As control targets, various vibrational states are selectively excited with laserpulses.
Chapter 2 supplies the necessary physical background for these studies. This mostly includesan introduction into the relevant parts of molecular physics, but also a primer on quantum controltheory. Chapter 3 follows up with a detailed description of the calculational methods central tothis work, i.e. MCTDH and CRAB. It also includes a description of the non-linear optimizationalgorithms which are used later. Chapter 4 then describes the performed calculations. Whilesome of these are mere test runs and may be omitted by a reader familiar with MCTDH, this thesis
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proceeds to increasingly relevant systems via the Morse oscillator to the LiCs molecule, whereexperimental potential data are used. The second half of chapter 4 includes optimal control runswith CRAB. Chapter 5 describes and discusses the results of these simulations, while section 6gives a conclusive summary.
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2 Physics of Molecules
This chapter will give a short overview of the molecular physics which are subject of this work.Exact analytical solutions are generally impossible for molecular systems, and even numericalapproaches often run into difficulties due to the high dimensionality. Therefore it is common toemploy some approximations, the most common being the Born-Oppenheimer approximation[8]. Section 2.1 describes this essential tool of molecular physics. It allows to describe elec-tronic and nuclear motions seperately, giving rise to electronic and vib-rotational eigenstates.Section 2.2 analyzes these eigenstates for the important special case of diatomic molecules.Having understood undisturbed molecules sufficiently well, it is time to study the interaction ofmolecules with external fields in section 2.3. Section 2.4 then discusses techniques to optimizesuch fields for a specific purpose, which is known as Quantum Optimal Control Theory (QOCT).Structure and contents of this chapter follow the relevant parts of [8] and [11] closely.
2.1 Born-Oppenheimer approximation
As a non-relativistic1 quantum system, a molecule obeys the Time-Dependent Schrödinger Equa-tion (TDSE) [8]
i~∂
∂tΨ(t) = HΨ(t) (2.1)
Here Ψ(t) denotes the molecule’s wavefunction and H is the Hamiltonian operator. Given Hand an initial value Ψ(t0), the system’s state is uniquely determined for all times. Assuming amolecule with Nn ≥ 2 nuclei and a total of Ne ≥ 1 electrons, the Hamiltonian can be specifiedand split into parts:
H = Tn + Te + Vn−n + Ve−e + Ve−n (2.2)
= −~2Nn∑i=1
∇2i
2Mi
− ~2Ne∑k=1
∇2k
2m
+e2
4πε0
Nn∑i<j
ZiZj
|Ri −Rj|+
e2
4πε0
Ne∑k<l
1
|rk − rl|− e2
4πε0
Nn∑i=1
Ne∑k=1
Zi
|Ri − rk|
1Throughout this work, relativistic effects will be neglected, which is common in computational chemistry withlight elements [12].
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The five terms correspond to the nuclear and electronic kinetic energy, the nuclear-nuclear,electron-electron and nuclear-electron potential energy, respectively. Mi is the mass of the ithnucleus, m the electron mass. ∇i is a gradient operator which only acts on the coordinates ofparticle i, which are denoted Ri for nuclei and ri for electrons [8].
Before proceeding, the notation can be simplified by introducing atomic units. In atomicunits (a.u.) e = ~ = m = 1
4πε0= 1, so most constants can be dropped from equation (2.2). The
natural unit of length in a.u. is the Bohr length defined by a0 = 4πε0me2
≈ 0.053 nm = 0.53 A(the radius of ground state hydrogen in the Bohr model), while energies are now measured inmultiples of a Hartree: 1Ha = ~2
ma20≈ 27.2 eV ≈ 2.1947 · 105 cm−1 (twice the binding energy
of a hydrogen atom) [11]. All quantities in this work are given in atomic units unless explicitlynoted otherwise.
For a time-independent Hamiltonian such as (2.2), the TDSE can be solved by finding thesystem’s eigenstates Ψn and expanding the wavefunction in this basis
Ψ(t) =∑n
cne−iEn/tΨn, (2.3)
where the En are the energy eigenvalues and fulfil the time-independent Schrödinger equation
(H − En)Ψn = 0. (2.4)
There are no analytical solutions of (2.4) for even the simplest molecules [8], but a number ofapproximations may be made to simplify the problem. The Born-Oppenheimer (BO) approxi-mation starts from the assumption that the electrons move much faster than the nuclei, so from anelectron’s point of view, the nuclear positions R are almost static. The electronic wavefunctionadapts to any nuclear configuration adiabatically, making the nuclear motion negligible. Hencewhen calculating the electron distribution, one may consider R to be constant for any giventime. In other words, the nuclear kinetic energy Tn is much smaller than the electronic energyTe+Ve−e+Ve−n, allowing for a perturbational approach to (2.4). The Hamiltonian may be splitinto an unperturbed part H0 = Te + V and the perturbation H1 = Tn. This leads to a clampednuclei Schrödinger equation:
H0φel(r;R) = E0
n(R)φeln (r;R) (2.5)
The semicolon2 indicates that the electronic wavefunction φeln (r;R) only takes the electron po-
sitions r as independent variable, whereas R is just a parameter [18]. Note that H0 only acts onr, too. Now assume (2.5) has been solved (either numerically or, in very simple cases as in H+
2 ,analytically) for an arbitrary fixedR such that the electronic eigenfunctions φel
n (r;R) are known.Without loss of generality, these eigenfunctions are orthonormal and can be used to expand the
2Some textbooks use the notation φel(R; r), φel(r|R) or φel(r,R) instead.
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full molecular wavefunction as
Ψ(r,R) =∑n
χn(R)φeln (r;R) (2.6)
It is worth remembering that Ψ is a function of both the electronic and the nuclear coordinatesr and R (hence the comma), and the nuclear wavefunctions χn depend only on R, of course.Inserting equation (2.6) into the molecular Schrödinger equation (2.4) eventually leads to twocoupled systems of equations for φ(r;R and χ(R):
H0φ = E0φ (2.7)
H1χn +∑m
(cnmχm) = (E − E0n)χn (2.8)
with the coupling coefficients
cnm :=
∫φ∗nH1φmdr−
1
2
∑k
(∫φ∗n
1
Mk
∂
∂Rk
φmdr
)∂
∂Rk
(2.9)
Up to this point no approximations have been made. The BO approximation now states thatall cnm = 0, completely uncoupling equation (2.7) from (2.8). In other words, the molecularwavefunction is assumed to factorize simply into
Ψ(r,R) = χ(R) · φel(r;R) (2.10)
This approximation can be justified by the fact that the coupling coefficients are very smallcompared to the H1 + E0
n term in equation (2.8) [8].Using the BO approximation, the problem of molecular dynamics can be solved in two steps:
First solve the electronic Schrödinger equation (2.5) assuming clamped nuclei for any positionR. In practice, this can usually only be done for a finite number of points, requiring interpo-lation for intermediate values. This gives the electronic energy levels E0
n(R) which dependparametrically on the nuclear positions. These energies can then be inserted into the nuclearSchrödinger equation, acting as a potential energy there. Now the reduced version of equa-tion (2.8), (Tn + E0
n)χn = Eχn can be solved for the nuclear wavefunctions, yielding the solu-tions χn,i(R). While n indexes the electronic energy levels, i counts the nuclear (i.e. vibrationaland rotational) energy levels for a fixed electronic state.
SinceE0n(R) can be interpreted as a potential in the nuclear Schrödinger equation, it is usually
called the Potential Energy Surface (PES) of state n and denoted by U(R). It includes all poten-tial terms from equation (2.2) and the averaged kinetic energy of the electrons. In the classicalpicture, the nuclei may be seen as masses moving on such a (typically multi-dimensional) PES.
A few words on the physical interpretation of the BO approximation: As the electrons movemuch faster than the nuclei, they adapt to any change of the nuclear position almost instanta-
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neously. This serves to justify both the clamped nuclei approach when calculating the electronicwavefunction and the inclusion of the averaged electronic energy into the PES.
According to the adiabatic theorem, the electronic wavefunction remains in the correspondingeigenstate φn during such a gradual change of H1 without coupling to other eigenstates φm [11].In other words, the molecule will always remain on the same PES.
However, the adiabatic theorem, and with it the BO approximation, breaks down at crossingsof PESs. More on this in section 2.5.
2.2 Eigenstates of diatomic molecules
Having separated the electronic from the nuclear wavefunction, both problems can be solved in-dependently. This section will at first turn towards the former, describing the electronic states ofmolecules. Then vibrational and rotational eigenstates are introduced. This also means climbingdown the energy scale from visible and UV light (exciting electronic transitions) over IR light(typical vibrational energies) down to the microwave regime (for rotational energies) [18].
The discussion will be restricted to diatomic molecules for two reasons: On one hand, onlydiatomics are subject of the studies in chapter 4. On the other hand, the energy structure ofdiatomics allows for a rather systematic and instructive description building upon many conceptsknown from atomic physics.
Although there is always coupling between the electronic, vibrational and rotational modes ofa molecule, most of these interactions are neglected in this work. Unless otherwise noted, a non-rotating molecule in the electronic ground state can be assumed. For the sake of completeness,the respective eigenstates will still be explained now.
Electronic states
Assuming a static frame of nuclear positions R, the electronic equation (2.5) can be solved.Again, analytic solutions are generally not feasible (the H+
2 ion being a notable exception), butfurther simplifications are possible. An important concept are molecular orbitals (MOs), whichtransfer the notion of electronic orbitals from atomic to molecular physics. Single electrons areassigned to specific states characterized by a set of quantum numbers, while the interaction withthe other electrons is either ignored or accounted for by mean fields. The total electronic wave-function then becomes a Slater determinant, i.e. an anti-symmetrized product of single-electronwavefunctions.
As in the atomic case, the eigenstates are conveniently classified by a set of quantum num-bers. Good quantum numbers are eigenvalues of operators which commute with the molecularHamiltonian [18]. For atoms, these include L2 (the total orbital angular momentum of the elec-trons squared) and Lz (the z component of the orbital angular momentum). When moving fromatoms to molecules, radial symmetry is lost, so total angular momentum is no longer conserved
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(in other words, L2 and H do not commute). The potential is axially symmetric, though, so Lz
still commutes with H .Its eigenvalues M are integers3, and molecular states are classified by the value Λ = |M | ∈
0, 1, 2, . . . . Capital Greek letters are used to denote different values, with Σ,Π,∆,Φ, . . . signi-fying Λ = 0, 1, 2, 3, . . . , respectively. States with non-zero Λ values are twofold degenerate, asM = n and M = −n correspond to the same energy.
The spin operators Sz and S2 still commute with the Hamiltonian, preserving S as a goodquantum number.
Each pair (Λ, S) defines an electronic term, a set of degenerate electronic states. This degen-eracy is expressed by adding the multiplicity 2S + 1 as a superscript to the Greek symbol. Forexample 1Σ denotes a singlet state with Λ = S = 0, while the triplet 3Σ is degenerate due toS = 1 (which permits the orientations Sz = 0,±1).
In molecules with point symmetry4, an additional subscript g or u denotes whether the wave-function is gerade or ungerade, i.e. whether it is invariant under inversion at the centre or changesits sign. And similarly, a Σ wavefunction may or may not change its sign upon reflection in aplane containing the molecular axis and is accordingly dressed with a plus or minus sign.
A full term symbol may thus be as simple as 3∆ for a molecule with Λ = 2, S = 1 or besomewhat complex like 1Σ+
g for Λ = 0, S = 0 with a wavefunction which does not change signunder inversion or reflection.
All terms can now be ordered by total energy. The ground term is denoted by the letter X ,while higher-energy terms are ordered alphabetically as A,B,. . .
Here capital letters are used for terms with the same multiplicity, while terms with a differentmultiplicity usually receive small letters (a,b,. . . ) [18].
Vibrational states
The focus of this work lies on vibrational states, as these are crucial for laser induced dissociationof molecules and the dynamics of many chemical reactions.
The solution of the electronic problem usually leads to the multi-dimensional PES U(R) of aparticular electronic state. In the diatomic case however, U(R) only depends on one variable, theinternuclear distance R. This allows to separate off the angular-dependent parts of the nuclearwavefunction χ(R) = ψ(R)L(θ, ϕ), reducing the problem to the one-dimensional Schrödingerequation. After switching to the centre-of-mass system and introducing the reduced massMr =
M1M2/(M1 +M2), the nuclear wavefunction obeys [8]:(− 1
2Mr
∂2
∂R2+ U(R)
)ψ(R) = Eψ(R) (2.11)
3This holds in atomic units. In SI units, the eigenvalues are M~.4In the diatomic case, this includes only homonuclear molecules.
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Figure 2.1: Comparison of a molecular potential and its quadratic approximation with respectiveenergy levels. Red (solid) line: Morse potential with parameters D = 0.2, β =1.5, R0 = 2.0. Green (dashed) line: Harmonic approximation around R0.
At least the electronic ground state provides a potential minimum at a distance R0, so the exis-tence of a stable ground state is guaranteed (otherwise no stable molecule can be formed). Theequilibrium distanceR0 may range from 0.74 A for H2 up to 52 A for the extremely unstable He2[7, 10]. Most diatomic molecules lie between one and a few A in size. For small energies, theinternuclear distance will oscillate around the equilibrium, resulting in a vibrating motion of thenuclei along the molecular axis.
Near the minimum, U(R) can be expanded as a Taylor series:
U(R0 + x) = U(R0) +1
2U ′′(R0) · x2 +O(x3) (2.12)
In some molecules the O(x3) term is negligible for the lowest energy states, so their spectrumresembles that of a harmonic oscillator. In particular, the lowest energy levels are almost equidis-tant. One example of an anharmonic potential is given by the Morse potential depicted in fig-ure 2.1. It is defined by
U(R) = D ·(1− e−β(R−R0)
)2 −D (2.13)
and has been used to model the potential of diatomics for a long time [8]. D is the dissociationenergy, R0 is the equilibrium distance and β is the width parameter of the potential. Close toR0, the quadratic approximation is quite accurate, and in the figure the first two energy levels
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almost coincide with the harmonic energies. With increasing energy and displacement fromequilibrium, the approximation breaks down. For large R, the Morse potential physical flattensand converges to zero5. The energetic distance between eigenstates decreases as dissociation isapproached, but only a finite number of bound states is supported.
For smallR, the Morse potential grows exponentially, effectively prohibiting the system fromreaching R = 0. In physical molecules, the potential actually diverges at the origin, but for aqualitative description the Morse model is often sufficient. It has the great advantage of beinganalytically solvable. The eigenenergies are
En = −D + ω
(n+
1
2
)− ω2
4D
(n+
1
2
)(2.14)
where ω = β√2D/Mr is the frequency of the corresponding harmonic oscillator and the sec-
ond term is the result of the anharmonicity of the potential. As in the harmonic oscillator, thevibrational quantum number n can take non-negative integer values, but here it has an upperlimit [8].
Other, more realistic molecular potentials have a different anharmonicity term, but usuallyshow comparable features: Convergence to zero for R → ∞, increasing density of states neardissociation and divergent behaviour for R → 0. Eigenfunctions resemble harmonic eigen-functions for small n, but look more and more like plane waves (at least for large R, where thepotential is almost zero) near dissociation.
As an example, the potential energy of the LiCs diatomic is shown in figure 2.2.
Rotational states
Apart from vibrations, molecules can perform rotational motions. A useful approximation isthe rigid rotor model, i.e. the assumption of two masses with a fixed distance R rotating aroundthe centre of mass. This is in agreement with the procedure of the last section, where the angularmotion had been separated from the vibrational motion along the molecular axis.
The rigid rotor is a standard textbook exercise [8] with the energy levels
EJ =J(J + 1)
2MrR2(2.15)
where the rotational quantum number J can take the values 0, 1, 2, . . . . The eigenfunctions arethe well-known spherical harmonics YJM(θ, ϕ). A diatomic molecule has axial symmetry, soonly two possible axes of rotation exist, both perpendicular to the molecular axis.
In reality, the internuclear bond is not rigid, but widens under the influence of centrifugalforce. As a result, the equilibrium distance increases with J , and with it the total energy (because
5The specific value is of course convention. It is common to assign negative energy values to bound states anddefine the zero of the energy scale to lie at R = ∞.
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Figure 2.2: Comparison of LiCs potential energy curve (red, solid line) with fitted Morse poten-tial (green, dashed line). Data from [24].
a larger nuclear distance amounts to an increase in potential energy). In terms of the angularmomentum |J| =
√J(J + 1) and the non-rotating equilibrium distance R0,
Erot =|J|2
2MrR20
− |J|4
2kM2rR
60
+|J|6
2k2M3rR
100
+ . . . (2.16)
Here k = − ∂∂RU(R = R0) is the spring constant of the potential energy curve. For a derivation
and more detailed discussion, including the effects of the electron angular momentum (omittedhere), see [8].
An important deviation from the rigid rotor is the coupling between rotational and vibrationaldegrees of freedom. As the vibrations are much faster (one or two orders of magnitude) than therotations, many vibrations occur during one rotational period. The resulting change in R affectsthe rotational energy, so there is a constant transfer of energy between rotational and vibrationalmodes.
The rotational energy also gives correction terms to the potential energy U(R), as it is pro-portional to 1
R2 for fixed J . Since these corrections are rather small, all rotational effects will beneglected in this thesis and a non-rotating diatomic will be assumed in the following chapters.
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2.3 Molecules in electromagnetic fields
Until now, the Hamiltonian has been time-independent, and a molecule prepared in an eigenstatewould always remain in that state. The total energy is conserved, too. In a non-static electro-magnetic field, this is no longer the case. The interaction with the field is modelled by adding anexplicitly time-dependent term to the Hamiltonian. This allows for transitions between states andchanges in energy via the absorption or emission of photons. More importantly, the moleculecan be driven into a desired state by an appropriately designed field. Lasers, providing a widerange of strong, coherent light fields in the visible to IR range, have become the most importanttool in quantum control of molecules [11].
Some approximations and assumptions
There are different ways to state the interaction between light and matter. This work will stickto the semi-classical approach, which treats the molecule as a quantum object, but the lightfield classically6. Furthermore the length gauge is used, making the interaction term particu-larly simple7. Since the wavelength of such lasers is on the order of at least several hundrednanometres, the dipole approximation is justified. It states that the electric field can be assumedto depend solely on time, not on position. For small molecules extending over little more thana few Angstrøm, this is pretty accurate [11]. In the dipole approximation, the electric field E ofan idealized (i.e. perfectly monochromatic) laser can be written as
E(t) = E0 · cos(ω · t+∆) (2.17)
Here∆ is a (usually irrelevant) phase, andE0 = E0·ε contains the amplitudeE0 and polarizationvector ε of the field8. Assuming linear polarization, one can choose coordinates such that the xaxis coincides with the polarization, projecting the last equation to one dimension.
Electric dipole moment
An electric field exerts no total force on a neutral molecule, but couples to it via its dipole mo-ment. When H0 = Tn +U(R) is the unperturbed Hamiltonian of nuclear motion, the dynamicsin the field E(t) are governed by the Hamiltonian [8]
H = H0 − µ · E(t) (2.18)
6A full quantum approach would require quantum electrodynamics and is not necessary at this point [18].7There is no need to dive into gauge theory here, so let it suffice to say that Maxwell’s equations are invariant
under a field transformation (Φ,A) 7→ (Φ− χ,A+∇χ) for any scalar function χ(r, t).χ can be chosen such that A vanishes and Φ = −r ·E, removing any A terms from the TDSE [11].
8Apologies for using the same letter E for energies and electric fields. Care was taken to explicitly state whichquantity is meant whenever the context is unclear.
17
Here µ is the transition dipole moment of the molecule. Its matrix elements can be calculatedby integrating the analogue of the classic dipole moment [11]
µfi(R) =
⟨φelf
∣∣∣∣∣Nn+Ne∑k=1
qk · rk
∣∣∣∣∣φeli
⟩el
(2.19)
The sum runs over all charged particles (nuclei and electrons) in the molecule, with the charge qkand position rk (for a moment, the convention to use r exclusively for electrons is abandoned).Integration in the scalar product is performed over the electronic coordinates. φel
i and φelf denote
the initial and final state of a transition. The off-diagonal elements f 6= i represent transitionsbetween different electronic states, while the diagonal elements f = i are responsible for vi-brational transitions on a single PES; the indices are usually dropped in the latter case. µ(R)
then denotes the expectation value of the dipole operator, or the permanent molecular dipolemoment9.
In the special case of a non-rotating diatomic molecule, this reduces to a one-dimensionalfunction of one variable, and the full Hamiltonian takes the form
H = − 1
2Mr
∂2
∂R2+ U(R)− µ(R) · E(t) (2.20)
The electric dipole moment has the dimensions charge · distance. Since the SI unit C·m isway too large to be useful in chemistry, it is often measured in atomic units (with 1e · a0 =
8.48 ·10−30C ·m) or Debye (1D = 3.34 ·10−30C ·m = 0.393 a.u.). Typical values for diatomicmolecules lie between 0 and 20 atomic units [8].
It is worth noting that a molecule has a non-zero permanent electric dipole moment if andonly if the centre of charge of the nuclei differs from the centre of charge of the electrons [8].Therefore molecules with inversion symmetry, such as CO2 or H2, have a vanishing dipole mo-ment and cannot be vibrationally excited with lasers. In the diatomic case, this rules out allhomonuclear molecules, so only heteronuclear molecules are accessible to IR spectroscopy.
The dipole moment µ(R) is usually a non-linear function of the position, although a linearapproximation may be reasonable in some cases. A popular model is the Mecke function [11]
µ(R) = µ0 ·R · e−R/R∗ (2.21)
with the parameter R∗ typically of the same order of magnitude as the internuclear distance.An example of such a function is plotted in figure 2.3. Close to the equilibrium distance, theapproximation
µ(R) ≈ −µ′ · (R−R0) + µ(R0) (2.22)
9To be distinguished from induced or temporary dipole moments, which are important for Van der Waals interac-tion between unpolar molecules, but shall not be dealt with here.
18
Figure 2.3: Mecke function with R∗ = 1.134, µ0 = 1.634 and linear approximation at R0 =1.821. This models the OH stretching mode in water [15]. Figure after [11].
with the dipole gradient µ′ = − dµdR
(R = R0) = µ0 · e−R0/R∗ · (R0/R∗ − 1) holds.
Rabi oscillations
As a simple example of light-matter interaction, consider an electron in a two-level system witha monochromatic electric field E(t) = E0 cos(ωt). The unperturbed eigenstates are |g〉 and |e〉with the energies Eg and Ee. The Hamiltonian is
H = Eg|g〉〈g|+ Ee|e〉〈e|+ Ω · cos(ωt) (2.23)
where Ω = µ12 · E0 = 〈e|r|g〉 · E0 is called the Rabi frequency. The Schrödinger equation canbe solved analytically with the linear combination ansatz |Ψ(t)〉 = cg(t) · |g〉+ ce(t) · |e〉, for adetailed calculation see e.g. [11].
In case of resonance, with i.e. ω = |Ee − Eg|, the system oscillates between the two stateswith the Rabi frequency [11]. Starting with full population of |g〉 at t = 0, the time-dependentpopulations are
pg(t) = |cg(t)|2 = cos2(Ωt
2
)(2.24)
pe(t) = |ce(t)|2 = sin2
(Ωt
2
)(2.25)
19
When the driving field is detuned by an amount ∆ = |Ee − Eg| − ω, the oscillation is fasterwith the frequency Ω′ =
√Ω2 +∆2. However no complete population transfer is possible with
a detuning, the maximum excited population is max(pe(t)) = (Ω/Ω′)2.Population transfer is a common goal in optimal control, so it is worth taking a closer look at
when it happens in a Rabi oscillator. Obviously pe = 1 is true for Ωt = π, 3π, 5π, . . .
In control experiments, laser pulses are often used rather than constant amplitude fields (cwlasers). The electric field is therefore modified to E(t) = E0 · f(t) · cos(ωt) with an envelopefunction f(t). Typical pulse shapes are Gaussian or sin2 functions. Now the area theorem can beshown to hold: It states that the population transfer only depends on the area under the envelopefunction, not on the form of f(t). As a corollary, complete population transfer occurs (resonanceassumed) whenever
∫ t
0f(τ)dτ = π. Such a pulse is accordingly called a π-pulse. Similarly, a
π2-pulse can be used to obtain a coherent superposition state 1√
2(|e〉+ |g〉) [11].
At first glance these results seem not very useful, as molecules are never true two-level sys-tems. But it turns out that many transitions can be effectively modelled this way if there areno neighbouring states with similar transition energies. In fact, even the resonant excitationfrom the ground state to the first vibrational state in a Morse oscillator can behave accordingto equation (2.23) because the off-resonant excitation to higher states is much weaker than the|0〉 → |1〉 transition. Since both states are rather localized around the equilibrium distance, eventhe linearised dipole moment of equation (2.22) can be used for good results.
2.4 Quantum Optimal Control
Now that the tools are available to calculate the evolution of a nuclear wavepacket under anexternal field, it is tempting to ask the inverse question: Given an initial and a target state, can amatching light pulse be found?
A more general statement of the problem reads: Given a fixed evolution time T , an initial stateΨ(t = 0) and a Hermitian operator O, what is the function E(t) that extremizes the expectationvalue |〈Ψ(T )|O|Ψ(T )〉|2 in the final state Ψ(T ) = e−iHTΨ(0)? In order to restrict the field tophysically meaningful and experimentally feasible values, some type of penalty term is addedto receive a cost functional which shall be minimized:
J [E] = 1− |〈Ψ(T )|O|Ψ(T )〉|2 + α ·∫ T
0
|E(t)|2dt (2.26)
The integral is proportional to the total pulse energy, multiplied by a suitable penalty factor α.It should be noted that other, more sophisticated penalty functions have been used which alsopunish high field gradients or undesirable pulse shapes [4, 5].
A common choice for O is the projection operator onto a specific target state, so the optimalcontrol problem becomes a matter of minimizing the infidelity I = 1− |〈Ψ(T )|ΨTarget〉|2 whilekeeping the field strength moderate [23].
20
Optimal control problems can be tackled both experimentally and theoretically. The experi-mental approach usually employs adaptive feedback control (AFC), which implements a closedloop design in the laboratory. The system under study is subjected to a shaped light pulse and itsresponse is measured. From this result, a learning algorithm calculates a modified pulse, whichis again fed into the pulse shaper. This process is repeated until – ideally – the control target isreached [4].
Although AFC is crucial for optimal control in practice and has produced great results, thisthesis will exclusively deal with the complementary approach, optimal control theory (OCT).It undoubtedly has its disadvantages compared to AFC: The wavefunction has to be propagatedwith some effort and under many approximations, while the molecule in the laboratory solvesits own Schrödinger equation in real time. Furthermore, the optimized pulses resulting fromtheoretical calculations are often hard to implement in reality.
Still OCT is justified: It can be much easier to set up a simulation with a model Hamiltonianthan conducting an experiment, especially for systems which are too unstable, noisy or exist onlyunder extreme conditions. The theoretically optimized pulse may not perform best in practice,but can usually give a good initial guess for AFC. Most importantly, calculations may give abetter understanding of the key processes driving the system, allowing for a more directed searchin the laboratory [4].
A number of optimal control algorithms exist to iteratively find an optimal pulse. As anillustrative example, consider the pump-dump scheme [16]. As a precursor to modern OCTmethods, the pump-dump scheme (also known as Tannor-Rice scheme) is very simply structured,but it makes the connection of equation (2.26) to laser selective chemistry clearer than other,more abstract methods [11].
Assume a chemical reaction with two possible outcomes: A molecule A2B can dissociate intoA2 + B or A+AB. Each channel corresponds to a valley in the ground state surface S0 (seefig. 2.4) leading to dissociation. The excited electronic state PES S1 has a roughly parabolicshape in this example. The molecule is then exposed to two consequent laser pulses. The pumppulse excites the system to the higher PES, lifting the nuclear wavefunction vertically (i.e. with-out change in the nuclear coordinates). The wavefunction is now generally not in a vibrationaleigenstate of S1, so it will propagate on the surface. After a while (typically tens of fs) the dumppulse de-excites the molecule back to the ground state. Depending on the delay between pumpand dump pulse, the wavepacket will end up in a different valley of the ground state PES andcontinue to propagate along this route. Hence with an appropriately chosen delay, the yield ofthe desired dissociation channel can be enhanced.
This technique, pioneered by D.J. Tannor and S.A. Rice [16], can be considered to be thefirst practical optimal control scheme, although in its very basic version described above, it isperformed with only one parameter. However, pulse shapes, phases and frequencies can also beoptimized to further increase the selectivity of the reaction.
Despite its appeal, the pump-dump scheme is very limited. Depending on the pulse length
21
Figure 2.4: Ground and excited state PESs of an A2B molecule. The arrows indicate possiblepump and dump pulses (image from [16]).
and the shape of the excited potential surface, dispersion can become a serious problem. Thewavepacket will then lose its (previously localized) shape, reducing the selectivity [16]. Further-more, only a few specific systems exhibit the necessary structure of potential energy surfaces,while most molecules are not suitable for the pump-dump scheme at all. Therefore, a more gen-eralized approach for OCT is needed, e.g. the Krotov method. It provides an iterative algorithmwhich produces successively better field guesses [11]. It is suitable (and has been used) fora wide range of quantum control problems; however only its application to quantum chemistryproblems is of interest here. The mathematics is rather tedious and not very relevant at this point,so the reader may consult Krotov’s comprehensive book on the subject[17] or the summary in[11] for details.
2.5 Non-adiabatic dynamics
Until now all coupling between electronic and nuclear motions has been neglected. This ap-proach will be pursued for the rest of this thesis, but its limits and the occasional breakdown ofthe BO approximation shall be discussed here.
In the purely electronic Schrödinger equation (2.5), the nuclear kinetic energy Tn was ne-glected as it is generally much smaller than the residual Hamiltonian H0 = V + Te In an ac-curate description, it must be treated as a perturbation λ ·W . In perturbation theory, the total
22
Hamiltonian is written as
H = H0 + λW (2.27)
where usually λ = (m/M)14 is used as the perturbation parameter [8]. When neglecting the
perturbation term, the unperturbed electronic eigenstates φeln and eigenenergies E(0)
n are found.Since λ 1, the actual energies can be found by means of a series expansion:
En = E(0)n +Wnn +
∑k 6=n
Wnk +Wkn
E(0)n − E
(0)k
+O(λ3) (2.28)
The third order terms are often neglected, since even for hydrogen (m/M)34 < 4 ·10−3. The first
and second order correction terms depend on the matrix elements of the perturbation operator,i.e. the nuclear kinetic energy:
Wnk = 〈φeln |Tn|φel
k 〉 =∫
(φeln )
∗Tnφelk d
3r (2.29)
These matrix elements describe the coupling between electronic states due to nuclear motion[8]. As long as the energy separation E(0)
n − E(0)k between these states is large compared to
Wnk, the coupling is weak and the expansion (2.28) holds. But when two electronic surfacesare close or even intersecting, the correction terms become large and the series diverges. Thisis when the BO approximation breaks down and the coupling becomes too strong to be treatedperturbatively.
In this case, even small changes in nuclear motion can trigger transitions between electronicstates. Effect such as radiationless decays occur, i.e. electronic transitions without emission ofa photon – the surplus energy is transferred into vibrational or rotational modes. Methods existto model molecular behaviour beyond the BO approximation, but they often go with greatercomputational effort [26].
The ground state surface is usually well separated from excited surfaces [8], so the BO ap-proximation is universally adopted for the rest of this thesis.
23
3 Numerical Methods
The purpose of this chapter is to make the reader familiar with the computational tools used inthis work. All propagation calculations in this work are performed with MCTDH (section 3.1),whereas CRAB (section 3.2) is being used for optimization of the external field. Since CRABis a general approach, a specific optimization algorithm has to be chosen. The ones used hereare the Nelder-Mead (or simplex) method and the Subplex method.
3.1 MCTDH
In this thesis the Multi-Configurational Time-Dependent Hartree method is the tool of choiceto simulate the time evolution of molecular systems. It is best understood by comparison withtwo competing schemes for propagating wavepackets [20]. Therefore sections 3.1.1 and 3.1.2will first introduce the numerically exact Full Configuration Interaction (Full-CI) and the ap-proximate time-dependent Hartree (TDH) method, respectively. In structure and notation, thissection closely follows the lecture notes [20] by MCTDH co-inventor HD Meyer, which shouldbe consulted for further comments and proofs.
3.1.1 Full-CI
The Full-CI approach (dubbed the standard method in [3]) is arguably the most straightforwardway of representing a multidimensional wavefunction. Consider a wavefunctionΨ(Q1, . . . , Qf , t)
depending on the f coordinates Q1 to Qf and on time. It may represent the nuclear positionson an f -dimensional PES, but the considerations below are fully general. It is now intuitive toexpand Ψ in a time-independent basis:
Ψ(Q1, . . . , Qf , t) =
N1∑j1=1
. . .
Nf∑jf=1
Cj1...jf (t)χj1(Q1) . . . χjf (Qf ) (3.1)
The functions χjκ(Qκ) can be any orthonormal basis, chosen according to the specific degree offreedom. For example, one might select Harmonic Oscillator (HO) eigenfunctions for a vibra-tional mode or Legendre functions for angular coordinates. Nκ then gives the basis size for thisdegree of freedom and must be chosen in a trade-off between the desired accuracy (the methodis exact for N → ∞) and computational effort.
24
In this representation, all time dependence lies in the coefficients Cj1...jf (t). Their equationsof motion can be found either directly by inserting the ansatz into the TDSE or equivalently witha variational principle, i.e. by demanding 〈δΨ|H − i ∂
∂t|Ψ〉 = 0 [20]. One finds [3]
iCL =∑J
〈χJ |H|χL〉CJ (3.2)
The multi-indices J = (j1, . . . , jf ) have the meaning χJ = χj1(Q1) . . . χjf (Qf ) (and similar forCJ ), while χJ is called a configuration.
The equation of motion (3.2) is a set of first order differential equations, which is formallysimple, but hard to solve in many practical cases. It involves N1N2 · · ·Nf coupled equations,and each Nκ should at least be about 10 (usually more) for good accuracy.
This exponential scaling makes Full-CI calculations impossible for anything but small sys-tems. Even massively parallel computing systems cannot handle more than six or seven degreesof freedom within reasonable time [20].
3.1.2 TDH
To circumvent the limitations of the Full-CI method, approximations have to be made. A verypopular and successful simplification is the TDH approach [8]. The ansatz is [20]:
Ψ(Q1, . . . , Qf , t) =
f∏i=1
a(t)φi(Qi, t)
Instead of a time-independent basis, TDH uses so-called single particle functions (SPFs)φi(Qi, t)
which are to be considered as wavefunctions of a single particle moving along coordinate Qκ.The wavefunction Ψ is approximated as a single product of SPFs, a so-called Hartree product.This greatly reduces the computational effort required for propagation.
Note that this representation is not unique, since arbitrary factors can be shifted between theSPFs and the prefactor a(t). In order to remove this redundancy, constraint functions have to beimposed. A general way to state the constraint functions gκ is
gκ(t) = i〈φκ|φκ〉 (3.3)
It is easy to show that all gκ have to be real in order to preserve the norms ‖φκ‖, but otherwisethey can be arbitrary functions of time [20]. These can be chosen in a way that simplifies theresulting equations of motion.
To state these equations, a little notation should be introduced. In analogy with the Full-CI
25
approach,
Φ =
f∏i=1
φi(Qi, t) (3.4)
is called a configuration. The single-hole configurations Φ(κ) are defined by
Φ(κ) =Φ
φκ
=∏i 6=κ
φi(Qi, t) (3.5)
By separating all variables Qκ, the TDH approach makes it necessary to account for particleinteraction in an averaged way. Here the mean field operators
H(κ) = 〈Φ(κ)|H|Φ(κ)〉 (3.6)
come in handy. Each of these operators acts only on the κth degree of freedom, as all othervariables have been integrated over. When acting on the SPF φκ, it effectively approximates theinfluence of all other particles by their averaged density.
This allows to state the equations of motion in the TDH approach:
ia =
(E −
f∑i=1
gi
)a
iφκ = (H(κ) − E + gκ)φκ
(3.7)
where the suggestive abbreviation E = 〈Φ|H|Φ〉 has been used. Note that all terms in equa-tion (3.7), including E, can depend on time.
It is now obvious why TDH is much faster than Full-CI: Instead of ∼ N f , now only N · fequations1 have to be solved [20].
Before doing so, the constraint functions must be specified. Apart from the simple and obviouschoice gκ(t) ≡ 0, an interesting result occurs for gκ(t) = E(t)/f . The equation of motion for athen simplifies to a = 0, resulting in
a(t) = a(0) = const.
iφκ =
(H(κ) − f − 1
f· E)φκ
(3.8)
This can easily be solved for tens of degrees of freedom with current technology. The maincomputational effort stems from calculating the mean-fields H(κ) at every time step, which bydefinition requires integrating over f − 1 dimensions. Fortunately, there are ways to reduce this
1The f + 1 equations of (3.7) can be reduced to f by suitable choice of gκ, see the following paragraph for anexample. For an actual calculation the SPFs still have to be expanded in a finite basis, here assumed to haveequal size N for all coordinates.
26
work, e.g. by approximating the mean-fields as a sum of lower-dimensional integrals.TDH’s computational efficiency comes at the cost of systematic errors. The separation ansatz
is exact if the Hamiltonian can be split into a sum H = Hsep = h(1)+. . .+h(f) where each h(κ) isa one-particle operator, i.e. only acts on the κth degree of freedom. This is usually true for kineticenergy operators, but not for most potentials. Here the Hamiltonian contains a non-separablepart H = Hsep+ V . The error between the exact propagation and the TDH result is proportionalto the difference between the true and the averaged potential (V −〈V 〉). Hence TDH is accuratewhen the potential changes only weakly over the width of the wavepacket. This condition issatisfied for well-localized wavefunctions in flat potentials. Notable errors are introduced whenthe wavefunction is widely spread or when the potential is very steep, resulting in large changesin V − 〈V 〉. This is often the case for higher vibrational states or whenever strong repulsionoccurs (e.g. when going to small R in a Morse potential) [20].
3.1.3 MCTDH
Although it has its merits, TDH is often too crude to give reliable results [8]. This is especiallytrue in correlated systems which are characterized by strong non-separable contributions to theHamiltonian [20, 3]. Full-CI computes all correlation effects correctly, but is too inefficient forlarger systems. The Multi-Configuration Time-Dependent Hartree method (MCTDH) attemptsto combine the advantages of both methods[19].
The wavefunction is expanded as a sum of Hartree products:
Ψ(Q1, . . . , Qf , t) =
n1∑j1=1
. . .
nf∑jf=1
Aj1...jf (t)ϕ(1)j1(Q1, t) . . . ϕ
(f)jf
(Qf , t) ≡∑J
AJ(t)ΦJ(Q, t)
(3.9)
In the last expression the multi-indices introduced in section 3.1.1 as well as the configurationsfrom equation (3.4) are used to shorten the multiple sums and products. This ansatz containsboth Full-CI and TDH as limiting cases: If n1 = . . . = nf = 1, the sum would reduce to onlyone configuration, yielding the TDH approach. If there were no time dependence on ΦJ , theexpression would be equivalent to the Full-CI representation with ∀κ : nκ = Nκ.
In practice it is useful to choose 1 < nκ < Nκ to include more than one configuration, butstill reduce the basis size compared to Full-CI. MCTDH can still give as accurate results with asmaller basis due to the time dependence of the basis functions ϕ(κ)
jκ. At each time step, they are
recalculated to optimally represent the current wavepacket, while the fixed basis of Full-CI maybecome inefficient.
As an example, consider a Gaussian wavepacket moving freely in a zero potential. The initialwavefunction (in arbitrary units) reads
Ψ(x, t = 0) = C · e−12x2+ip·x (3.10)
27
Figure 3.1: Spread of a Gaussian wavepacket. Thin red (solid) line: Initial wavefunction withmomentum p = 5 and initial width ∆x0 = 1. Strong red (solid) line: Wavefunction(absolute value) at t = 0.2. Blue (dotted) line: Expansion in terms of HO functions(center and frequency according to initial wavefunction) to fifth order.
with the initial position at x = 0, the momentum p and a normalization constant C. One wouldtypically choose a HO basis to represent such a wavefunction. Since the harmonic ground stateis a Gaussian, just one basis function (with a suitable width) is already sufficient for an exactdescription.
As Ψ evolves, it stays in Gaussian shape, but its width increases proportional to√1 + 4t2 and
it is centred around p · t (see figure 3.1) [1]. Such moving and possibly dispersing wavepacketsare characteristic of the dissociation of molecules [20]. Now the fixed basis of Full-CI quicklybecomes impractical, while a time-dependent basis of shifted and rescaled HO eigenfunctionsremains efficient. One advantage of MCTDH is to always use such an optimal basis set. Thisalso applies to non-Gaussian wavepackets.
When compared to TDH, MCTDH performs better since it accounts for correlation effects.With a good choice of configurations, all important correlations are covered. Little to no mem-ory is wasted on weakly or uncorrelated modes as in Full-CI, which by definition considers allconfigurations and possible correlations. The performance of MCTDH crucially depends on thechoice configurations [3].
After this prelude, it is time to introduce some more notation before turning towards the equa-tions of motion. Similar to TDH, there will be equations to propagate not only the coefficient
28
vector2, but also the SPFs. Another common feature is the need for constraints because therepresentation (3.9) is not unique. Not only can time dependent factors be shifted between theA-vector and the configurations ΦJ , but also changes in the SPF basis are possible. Any linearmap U (κ) : Rnκ → Rnκ can be performed on the κth SPF basis to obtain the transformed SPFs
ϕ(κ) = U (κ)ϕ(κ) (3.11)
where ϕ(κ) should be read as a vector (ϕ(κ)1 , . . . , ϕ
(κ)nκ ). When simultaneously A is subjected to
the inverse transformation(s)
AJ =∑L
AL
(U (1)
)−1 · · ·(U (f)
)−1 (3.12)
the wavefunction Ψ remains invariant. Instead of the constraint functions of TDH, MCTDHrequires constraint operators g(κ). The simplest way to define them is
i〈ϕ(κ)l |ϕ(κ)
j 〉 = 〈ϕ(κ)l |g(κ)|ϕ(κ)
j 〉 ≡ g(κ)lj (3.13)
for all κ = 1, . . . , f . Just as a real constraint function was chosen earlier to conserve the norm,hermitian constraint operators (i.e. g(κ) = g(κ)†) ensure orthonormality of the ϕ(κ)
l (as long asthis was given at t = 0).
A generalization of the previously used multi-indices is now introduced. As a reminder, amulti-index J is shorthand for (j1, . . . , jf ). Now let Jκ be a tuple of indices with the κth entryomitted, and let Jκ
l denote a multi-index with the κth entry replaced by l. In equations:
Jκ = (j1, . . . , jκ−1, jκ+1, . . . , jf ) (3.14)
Jκl = (j1, . . . , jκ−1, l, jκ+1, . . . , jf ) (3.15)
(3.16)
In this notation, the single-hole configurations from section 3.1.2 are written as
ΦJκ =∏ν 6=κ
ϕ(ν)jν
=ΦJ
ϕ(κ)jκ
(3.17)
This also allows to define single-hole functions
Ψ(κ)l =
∑Jκ
AJκlΦJκ (3.18)
Ψ(κ)l is a function of f − 1 coordinates, with all SPFs which on Qκ ruled out from Ψ. Thanks to
2Technically, A = Aj1...jf is not a vector, but a rank f tensor. Still the term A-vector is used throughout thereference [3] and consequently here, too.
29
the completeness of the SPF bases, the identity Ψ =∑nκ
l=1 ϕ(κ)l Ψ
(κ)l holds.
Using the single-hole functions, the mean field operators
〈H〉(κ)jl =⟨Ψ
(κ)j
∣∣∣H∣∣∣Ψ(κ)l
⟩(3.19)
and the density matrix
ρ(κ)kl =
⟨Ψ
(κ)k
∣∣∣ Ψ(κ)l
⟩=∑Jκ
A∗JκkAJκ
l(3.20)
can be defined. Again, the mean fields only act on the κth degree of freedom after everythingelse has been integrated over.
The same applies to the MCTDH density matrix, which should not be confused with thecommon definition of a density operator [20, 3]. The latter is ρ = |Ψ〉〈Ψ| for a pure state andoperates on all degrees of freedom. Only after tracing out all other degrees of freedom3, thereduced density is found which coincides with the definition (3.20) above.
Remembering the end of section 3.1.2, the Hamiltonian can be split into a separable and aresidual part H =
∑fi=1 h
(i)+ HR, where the h(κ) are one-particle operators. Since the effect ofthe separable part is easy to evaluate, only the residual Hamiltonian needs to be approximatedby mean fields:
H(κ) = 〈HR〉(κ) (3.21)
Finally, after introducing the projector onto the subspace of functions of the Qκ coordinate
P (κ) =nκ∑j=1
|ϕ(κ)j 〉〈ϕ(κ)
j | (3.22)
all tools are set up to state the equations of motion. They are found with the Dirac-Frenkelvariational principle, i.e. by taking the functional derivatives of Ψ with respect to the coefficientsAJ , the SPFs ϕ(κ) and time and then demanding⟨
δΨ
∣∣∣∣H − i∂
∂t
∣∣∣∣Ψ⟩ = 0 (3.23)
This leads to separate equations of motion for AJ and ϕ(κ) which still depend on the chosenconstraint operators. The full calculation can be found in [20], at this point just the results for
3The trace of a density matrix is∑
n〈φn|ρ|φn〉 where the |φn〉 form a complete orthonormal basis. In this casethe SPFs of the desired degrees of freedom provide such a basis, and a trace is performed over all SPFs exactthose for DOF κ.
30
g(κ) = 0 shall be stated. These are the full MCTDH equations of motion [20]
iAJ =∑L
〈ΦJ |H|ΦL〉AL
iϕ(κ)j =
(1− P (κ)
) [h(κ)ϕ
(κ)j +
nκ∑k=1
nκ∑l=1
(ρ(κ))−1
jkH(κ)
lk ϕ(κ)l
] (3.24)
As soon as the separable and the residual (correlated) part of the Hamiltonian have beenidentified and the initial SPFs have been defined, these equations can be used to start calculationsright away. Any numerical integrator such as Runge-Kutta can be used to solve the differentialequations. Some refinements to the algorithm exist to further reduce CPU time and memoryrequirements, most notably mode combination (aggregating several physical coordinates intoone logical coordinate Qκ = (q1, . . . , qk) with only one SPF ϕ(Qκ)) and Constant Mean Fields(saving time by updating the mean fields not at every time step). Although they can increase theperformance of MCTDH dramatically, the interested reader is referred to the lecture notes [20]for a thorough description.
The limiting factor in calculations is often the memory consumption, so it is interesting tocompare MCTDH with Full-CI here. The latter has to store a large coefficient vector CJ withN f entries4, and memory consumption is proportional to this number.
Plain MCTDH stores a number of variables proportional to nf + f · n · N where the firstterm is the number of entries in the A-vector and the second term accounts for the SPFs. Someextra space is required for the mean fields etc., but the overall scaling does not change. Sincethe number n of SPFs is usually much smaller than the grid size N , the exponential part growsmuch faster in Full-CI than in MCTDH, resulting in a gain of (N/n)f [20].
As a typical example, let N = 32 and n = 7. For four degrees of freedom, Full-CI requiresabout 48 MB of memory and MCTDH just 620 kB. The difference becomes even greater formore degrees of freedom. For f = 9, Full-CI would need about 1.5 PB which is absolutelyimpossible, while MCTDH still works with less than 8 GB. With mode combination, over thirtydegrees of freedom have been handled [3].
In terms of CPU time, MCTDH can be shown to beat Full-CI by a similar factor of roughly(N/n)f+1.
The Heidelberg MCTDH package
There are several implementations of the MCTDH scheme, but the most prominent one has beendeveloped by scientists of the Universität Heidelberg [25, 3, 19]. After more than 15 years ofdevelopment, the Heidelberg package now comprises a complete toolkit for multi-dimensionalquantum dynamics including wavefunction propagation, diagonalization of Hamiltonians, manyanalysis tools and more. Originally intended for quantum chemistry, it has also been applied
4Again assuming that all degrees of freedom require the same basis sizeN1 = . . . = Nf ≡ N . Similarly, identicalnumber of SPFs nκ ≡ n for all degrees of freedom is assumed in MCTDH.
31
to other problems where multi-dimensional wavefunctions appear, e.g. cold atoms or certainbosonic systems5. Even an optimal control tool named optcntrl was added by M. Schröder,implementing the Krotov and the Rabitz optimization schemes [23].
One advantage of the software is its ease of use. The Hamiltonian, initial wavefunctionand all parameters are provided via plain text input files. However the Hamiltonian must begiven as a sum of products of single-particle operators (i.e. a sum of separable parts) as inH =
∑sk=1
∏fi=1 h
(i)k to work with the algorithm. This is usually not a restriction for ki-
netic energy operators, and many simple potentials can also be written in product form, e.g.V (x, y) = v1(x)v2(y). Such a representation is often not given for a physical PES, but there isa solution: The Heidelberg package comes with POTFIT, a program to expand a given potentialin sum-of-products form.
In this thesis the Heidelberg MCTDH package will be consistently used for all propagationcalculations, specifically version 8.4.9. Therefore the name MCTDH is from now on used syn-onymously with this specific software. The program optcntrl is only used for comparison atsome points, since the major part of optimization is performed with CRAB.
One important, final remark on the calculations: Since all systems treated in chapter 4 areone-dimensional, the calculations are not in fact performed with the MCTDH method, as therepresentation 3.9 is just not meaningful for f = 1. Still the Heidelberg package was used,hoping MCTDH will play out its strengths when turning towards higher-dimensional systemsat a later point (but which are way beyond the scope of a Master’s thesis). This thesis is thus amere starting point for further research.
3.2 CRAB
Chopped Random Basis (CRAB) is a general-purpose algorithm for quantum optimal control.It was introduced by T. Caneva et al. [5] in 2011 and has been shown to be efficient for qubitmanipulation, spin coupling and other systems. In this work CRAB will be used to optimize thepulse parameters for state selective excitation, while MCTDH computes the system’s responseto the pulse.
The basic concept of CRAB is summarized easily: The control field (i.e. the light pulse) isexpanded in a finite set of randomized basis functions. The expansion coefficients are consideredas variables of the cost function which must be minimized.
To be more specific, consider a system subject to the control field Γ(t) for a fixed time T . Fornow one field shall be sufficient, though generalization to a tuple of fields ~Γ is straightforward.Just as before (equation (2.26)), a cost functional can be defined
J [Γ(t)] = 1− |〈Ψ(T )|ΨTarget〉|2 + α ·∫ T
0
|Γ(t)|2dt (3.25)
5The code was extended to include the correct symmetry relations, resulting in the software MCTDHB.
32
The first part measures the infidelity, while the last term minimizes the energy content of thefield. Quite typical of control algorithms, CRAB works best when supplied with a good initialguess Γguess(t) for the field. An iterative procedure is then used to optimize the corrected field
Γ(t) = Γ0(t) · g(t) (3.26)
with the correction g expanded in an orthonormal basis:
g(t) =∑k
ckχk(Ωk, t) (3.27)
The χk may be any suitable set of functions which depend on parameters Ωk, e.g. a Fourier basisor associated Laguerre polynomials6.
As already mentioned, CRAB relies on truncating (chopping) the basis to a finite size Nc.Now the basis is randomized by substituting the parameters Ωk 7→ Ωk = Ωk · (1 + rk) withrandom numbers rk. So the CRAB ansatz for g is
g(t) =Nc∑k=1
ckχk(Ωk, t) (3.28)
The orthonormality of the basis is lost in the process, but the search space is enlarged. Therefore,the randomization may speed up the convergence of optimization without loss of accuracy. Theauthors of [5] report results of comparable quality in fewer iterations.
Applying the above to an electric field in a Fourier basis, one arrives at
E(t) = Eguess(t) ·
(1 + Λ(t)
Nc∑k=1
(an sin(ωkt) + bn cos(ωkt)
)(3.29)
Here Λ(t) is a function with Λ(0) = Λ(T ) = 0, ensuring a smooth pulse shape. A commonchoice is sin2(πt/T ), which will also be used here. The frequencies ωk can be harmonic fre-quencies 2πk/T or characteristic frequencies of the system at question, usually multiplied witha random factor (1 + rk) ∈ [0.5, 1.5]. Often three to five frequencies are enough to give goodresults. The algorithm aims to optimize the 2Nc coefficients ~a and ~b until a minimum of J isfound.
Now the optimization problem has been reduced to minimizing a function f(~a,~b) = J [Γ(~a,~b; t)]
with respect to the parameters ~a,~b. The minimization of a multivariate function f : R2Nc → Ris a standard problem of applied mathematics, and a plethora of algorithms exist for the pur-pose of finding such a minimum. Many of them are so-called gradient methods, meaning theyneed the derivative of f either analytically or numerically. This is somewhat impractical for theproblems in this thesis, since an analytical expression for the derivative is not known and ap-
6The associated Laguerre polynomials Lαk (x) are the solutions of 0 = xy′′ + (α+1− x)y′ + ky, where α is any
real parameter.
33
proximate methods show a poor performance. The following section will therefore present twonon-gradient algorithms which are simple, robust and do not require any knowledge of deriva-tives [5].
3.3 Nelder-Mead and Subplex methods
The quantum control problem has now been wrapped into an minimization problem for the ob-jective function7 f(~c). This task will now be handed over to one of two specific algorithms.Both of them are direct search methods, i.e. they work by comparing function values at differ-ent points ck ∈ Rn rather than evaluating gradients or functional derivatives. This makes thisapproach very versatile and insensitive to noisy objective functions.
Nelder-Mead
The Nelder-Mead simplex (NMS) method was developed by J. Nelder and R. Mead in 1965[21] and is one of the most popular gradient-free search methods. It relies on a simplex movingthrough the n-dimensional search space and contracting when it has surrounded a minimum.Simplices generalize the concept of a triangle in 2D to higher dimensions [22]. An n-simplex isan n-dimensional convex polytope, i.e. it consists of a set of n+ 1 affinely independent8 points~xi ∈ Rn and all the points lying inside. The NMS algorithm proceeds as follows [22]:
1. Given a start value ~a ∈ Rn, construct the initial simplex with ~x0 = ~a. The other verticesare usually found by moving a certain distance along the respective axis.
2. Sort the function values f(~xi) by size and find the vertices with the highest function valueat ~xh and the second highest at ~xs. The currently best vertex (i.e. the one with lowest f )is named ~xl.
3. Calculate the centroid ~c as the arithmetic mean of all vertices except ~xh and the reflection~xr = 2~xc − ~xh of ~xh at the centroid. Now check ether ~xr is an improvement over thepreviously worst vertex.
4. If f(~xr) is lower than f(~xs), replace ~xh by the reflected vertex. If it is even the best vertex(i.e. f(~xr) < f(~xl) is a new minimum), try to expand in this direction by setting a newvertex at ~xr − ~c. Either way, return to step 2 with the new simplex.
5. If f(~xr) is greater than f(~xh), then the current simplex is likely to already contain theminimum. Contract it by setting a new vertex halfway between ~xh and ~c. If f(~xr) lies
7For simplicity, the two parameter vectors of the last paragraph have been combined into one vector ~c =(a1, . . . , aNc
, b1, . . . , bNc) of length n = 2Nc.
8That is, the connection vectors ~xi − ~xj for i 6= j form a linearly independent family. In simpler words, thiscondition makes sure that the polytope is really n-dimensional by excluding “triangles” consisting of just threepoints on a line.
34
between the highest and the second-highest function value, place the new vertex halfwaybetween ~xr and ~c instead.
6. If the new vertex from step 5 is better than both ~xh and ~xr, contraction was successful.Replace ~xh with the new point. Otherwise, shrink the entire simplex by moving all verticeshalfway towards ~xl. Return to step 2.
This procedure can be slightly generalized by introducing scaling factors e.g. at the reflectionor contraction steps. The above description corresponds to the most common settings. Thealgorithm usually halts when the objective function no longer changes significantly over thesimplex points, but a termination condition based on simplex size can also be used (and will bebelow).
NMS has the advantages of being rather robust to noisy functions and performs well for smalln (. 5). However, it quickly becomes inefficient for very large n and is badly suited for searchesincluding constraints. Convergence may fail completely when the simplex collapses into a sub-space [22].
As for most direct search methods, little theory exists describing the convergence propertiesof NMS. The popularity of NMS is founded on its success in practice.
Subplex
Dissatisfied with the poor performance of NMS for large n, T. Rowan developed an improvedalgorithm in his PhD thesis [22]. Subplex9 is short for subspace-searching simplex. The algo-rithm breaks down the search space into low-dimensional subspaces which can then be searchedby NMS. This trick can vastly speed up the calculation compared to NMS, but it risks neglectingsubspaces with even lower minima. As another advantage, the memory requirements of Subplexare O(n) instead of the O(n2) of NMS10.
For each iteration, Subplex divides the search space based on the so-called vector of progress,that is the change in ~xl during the last step. The subspaces are chosen such that the vectorof progress lies approximately in the first subspace; here the minimum is suspected. Furthersubspaces are generated by other elements of the vector of progress, grouped together by theirorder of magnitude. All subspaces are then consecutively searched by NMS, which halts whenthe simplex size has been reduced by a predefined parameter11. Subplex uses the new minimumfound by NMS to define the new vector of progress, update the subspaces and adapt the stepsizes. Although somewhat heuristic, Subplex has shown to work well in practice and often evenbetter than NMS [22, 13].
9The author preferred the capitalization SUBPLEX in his thesis. This is be abandoned here for readability.10This is in fact not crucial for any of the applications in this thesis, where n never exceeds 10. The computational
cost is dominated by the function calls (i.e. propagations with MCTDH), not by the search algorithm.11 1
10 was Rowan’s standard setting. Smaller values increase speed even further, but makes missing a better mini-mum more likely. Higher values improve search quality while slowing Subplex down.
35
Rowan wrote his original code in Fortran. This thesis uses S.G. Johnson’s implementation ofSubplex in C contained in the NLopt optimization library [13]. NLopt provides an interface forsimple integration with various programming languages; it also contains an implementation ofNMS used here.
36
4 Calculations
This chapter describes the calculations performed with MCTDH1, it is divided into three sec-tions. Section 4.1 contains basic exercises with a Harmonic oscillator, mainly to test the func-tions and settings of MCTDH. A reader already familiar with the software may skip this part. Insection 4.2 a more realistic system is investigated, the one-dimensional Morse oscillator. Thisserves as a simple, but qualitatively sufficient model of a diatomic molecule. The potential pa-rameters are taken from a paper by M. V. Korolkov et al. [15] whose results shall be reproduced.Finally, section 4.3 investigates this thesis’ central issue, the value of MCTDH for optimal con-trol problems. Here, the built-in optimization tool optcntrl is compared to a Fourier-expandedpulse as in equation (3.29) either with or without a randomized basis (CRAB). The two systemsunder study are the Morse oscillator just described and the LiCs diatomic with an experimentallydetermined potential.
Before starting any calculations in MCTDH, a number of simulation parameters have to beset. The most important are listed below. MCTDH reads all these parameters from two (or more)text files. All physical properties of the system are written into the operator file. This includesthe Hamiltonian and all physical constants, and possibly further operators which may be used.All computational parameters and settings are given via the input file, with the most importantones listed below.
• The discrete variable representation (DVR) has to be chosen. Whenever a real-valuedfunction is represented on a finite-size computer, a DVR is required [3]. It can be repre-sented in a position space basis by storing the function values at the grid points. Depend-ing on the type of function, an expansion in a function basis is often more efficient. Forexample, the wavefunction of an oscillating system can be expanded in Harmonic oscil-lator eigenfunctions. Both the type and the size (i.e. the number of grid points or basisfunctions) of the DVR are decisive for the efficiency and accuracy of a calculation.
• The number of SPFs must be defined. In multi-dimensional systems, it is crucial to includethe configurations with strong interactions and to skip those with negligible correlation.In the one-dimensional case here, this step is moot. There is only one variable and henceone SPF.
• The numerical integrator actually solves the MCTDH working equations (3.24). MCTDHoffers a number of algorithms, including the Bulirsch-Stoer (BS) integrator and the Short
1The software package, not the algorithm.
37
INPUT → MCTDH → OUTPUT → ANALYSIS
• input file
• operator
file
• (optional)data files
• output
file
• psi file
• gridpop
file
• . . .
• Plot data
• Energies
• Accuracy
• . . .
Figure 4.1: Workflow with MCTDH, from left to right: All parameters and settings are definedin the operator and input files. Additional data files can provide e.g. PES data or theinitial wavefunction. During the calculation, the program writes the desired outputfiles, which can then be plotted and analysed with included tools or external scripts.There are also tools to check the accuracy and efficiency of the calculation.
Iterated Lanczos (SIL) algorithm. Depending on the chosen method, some more parame-ters may be set. For most cases, the default settings (BS for propagating the SPFs, SIL forthe A-vector) produce reliable results.
• The initial wavefunction has to be set. There are several ways to do this, such as definingit as the eigenfunction of a given operator, by using a number of predefined functions (e.g.HO eigenstates), reading a numerical file or using the result of a previous calculation.
• The input file also specifies to what extent MCTDH shall save the output of its calculations.For example, the full wavefunction is often interesting, but may become prohibitively largeto store. In such cases, it may suffice to save just the grid populations or the autocorrelationfile. In total, MCTDH can write up to ∼ 20 different output files, and it would be wastefulto save everything indifferently.
While a well-written operator file can be reused many times for the same system, the input fileis usually adapted slightly to the aim of the current calculation. The workflow of an MCTDHcalculation is sketched in figure 4.1.
4.1 Harmonic Oscillator
The quantum harmonic oscillator (HO) in one dimension is defined by the Hamiltonian
H = − 1
2m
∂2
∂x2+
1
2mω2x2 (4.1)
38
Figure 4.2: Typical initial wavefunctions for the HO calculations. Black (thin) curve: harmonicpotential. Blue bell curves: Ground state (centralized) and coherent state (displaced).Green (wide) bell curve: Squeezed coherent state.
with the frequency ω and mass m. Its eigenenergies are the equidistant levels En = ω(n +12), n = 0, 1, . . . and its eigenstates are the known functions
ψn(x) = Nn · e−mω2
x2 ·Hn(√mωx) (4.2)
with a normalization constant Nn to ensure∫|ψn(x)|2dx = 1 and the Hermite polynomials2
Hn(x) [8].This system is extremely well-studied and allows for an analytical solution in many cases.
Therefore it is ideal for testing purposes, since one usually knows what to expect.
4.1.1 Propagation
The main purpose of MCTDH is the propagation of wavefunctions. Different initial wavefunc-tions were chosen and propagated in the potential with m = 1.0 and ω = 0.015198. Thiscorresponds to an oscillator period of 10 fs and an energy of 0.4136 eV. As a DVR, HO eigen-functions are used with a grid size of 24. The initial wavefunctions include (see also figure 4.2):
• HO eigenfunctions. These should stay in the respective eigenstate and show no evolution(except for an irrelevant phase).
• Displaced ground state wavefunction, also known as a coherent state. This is a Gaussianfunction with the same width as the ground state, but centre at some x0 6= 0. It should
2The Hermite polynomials are defined by Hn(x) = (−1)n exp(x2) dn
dxn (exp(−x2)). The first few functions areH0(x) = 1,H1(x) = 2x,H2(x) = 4x2 − 2, . . .
39
show a classic oscillatory motion around zero while keeping its shape. The same holdsfor coherent states with an initial momentum.
• Gaussian with an arbitrary width centred around zero. This is called a squeezed coherentstate. It should show a breathing motion, i.e. a periodic change in width. Any Gaussianwavefunction can be seen as the ground state of a HO with a specific frequency. In thiscase, this frequency is Ω = 0.01976, yielding a narrower curve than the ground state ofthe actual potential.
4.1.2 Relaxation
Relaxation is a technique for finding the ground state wavefunction of a given Hamiltonian bypropagating the initial wavefunction in negative imaginary time. For large t > 0, the wavefunc-tion Ψ(τ) ≡ Ψ(−it) converges to the ground state [20].
This calculation is performed with two-dimensional isotropic HO with m = 1 and ωx =
ωy = 8.0, the initial wavefunction is picked somewhat arbitrarily as a two-dimensional Gaus-sian centred around (x, y) = (0.6,−0.4) with width (∆x,∆y) = (2.08, 1.18), although therelaxation should converge to the same result for any reasonable input. The result is judged bythe final energy which should be E0 =
12(ωx + ωy) = 8.0 = 217.69108 eV. When the accuracy
is unsatisfactory, the primitive basis or integrator settings can be adjusted.
4.1.3 Coupling to a field
The Hamiltonian including field interaction reads
H = − 1
2m
∂2
∂x2+
1
2mω2x2 − µ0 · x · E(t) (4.3)
with the dipole moment set to µ0 = 1.0. The initial wavefunction (ground state or first andsecond excited states) is propagated with the field E(t) chosen to be
• a resonant monochromatic field E0 · cos(ωt) with ω being the oscillator frequency and theamplitude E0 ranging up to 0.15.
• an off-resonant monochromatic field E0 · cos(ωLt) with ωL 6= ω and comparable ampli-tude.
• laser pulses E0 · f(t) cos(ωLt) with the envelope function f(t) = cosh−2( t−∆b). Here
∆ = 4 fs sets the centre of the pulse; the width parameter b as well as the frequency ωL
are varied. Some examples are depicted in figure 4.3.
40
Figure 4.3: Four sample pulses used to excite the Harmonic oscillator.
41
4.2 Morse Oscillator
Not only is the Harmonic Oscillator unfit to model a diatomic molecule, it also shows limiteddynamics. An interaction as in equation (4.3) can only excite the HO to coherent states [9], neverto a single eigenstate |n〉. In order to study selective excitation, a non-harmonic system has tobe chosen such as the one-dimensional Morse oscillator introduced in section 2.2. Here theparameters of the potential (2.13) are the depth D = 0.1994, equilibrium distance r0 = 1.821
and the width parameter β = 1.189. These values are taken from [15], where this system isstudied in detail. It models the OH bond in water, so the reduced mass ism = (mHmO)/(mH +
mO) = 1728.539. The harmonic frequency (the frequency of the harmonic approximation nearr0) is ω = β
√2D/m = 0.01806.3 There are exactly 22 vibrational eigenstates from |0〉 to |21〉.
The authors of [15] used three consecutive IR pulses to excite the bond from the ground stateto |15〉, which is close to the dissociation threshold. The goal of this section is to reproducetheir result, taking one step at a time. The first step is to set up a suitable DVR and check theaccuracy of MCTDH. Then, population transfer between |0〉 and |1〉 via Rabi oscillations can beattempted. Having achieved that, the IR pulses from [15] can be modelled to excite the systemfirst to |5〉, then to |10〉 and finally to |15〉.
4.2.1 Choice of DVR
Although the lower vibrational states can be expanded quite well in terms of HO eigenfunctions,such a DVR fails for the higher eigenstates. As was mentioned before, only the low eigenstatesare sufficiently localized. States close to dissociation can have significant values far away fromr0 and have more resemblance to plane waves. Therefore a sine DVR is a better suited for thedissociative modes. MCTDH implements such a primitive basis with the basis functions
χj(x) =
√2
L· sin
(x− x0L
πj
)· 1[x0,xN+1](x), j = 1, 2, . . . (4.4)
where 1[x0,xN+1] is the indicator function of the interval [x0, xN+1] (i.e. it is one inside includingthe boundaries and zero outside). These are the eigenfunctions of a particle in a box with lengthL = xN+1−x0. By choosing the numberN of basis functions, the user divides the interval intoN equidistant grid points x1 to xN [20].
Based upon the results of the next section (4.2.2), a sine DVR with x1 = 0.7, xN = 8.0, N =
96 is chosen for most calculations. The grid boundaries can be explained when considering thevalue of the potential at these points: V (0.7) = +7.05 is high enough to prevent the wave-function from penetrating any further to the left. Smaller x1 turned out to lead to numericalproblems, presumably because the energy range becomes too large for floating point accuracy.Similarly, even the highest bound state has negligible probability density beyond of x = 8.0,
3In more common units, D = 5.43 eV, r0 = 0.96 A, β = 0.63 A−1, m = 1.055AMU and ~ω = 0.491 eV=3963.7 cm−1.
42
because V (8.0) = −0.000049 is still considerably larger than E21 = −0.00377. Higher xN isagain impractical since a very large basis size would be required to keep the spatial grid fineenough.
Fewer basis functions than 96 are usually enough, in factN can be reduced to 64 without lossof accuracy for all but the very last excitation step (last paragraph of section 4.2.3). However96 is used for all OCT simulations to be on the safe side and because the calculations are stillreasonably fast.
4.2.2 Diagonalization
MCTDH includes a feature to diagonalize an arbitrary hermitian operator with the Lanczosalgorithm [3]. This is applied to the Morse oscillator with the parameters specified above (4.2)to find the eigenenergies. Since these are known exactly by equation (2.14), this can serve asa benchmark for the DVR. Instead of a final propagation time, this type of calculation asks theuser for the number of iterations, typically about 103 to 104 for small systems. In this case, 5000steps are chosen, as higher numbers show no improvement in the results.
The diagonalization is performed for two types of primitive basis sets: A sine DVR with theend points 0.7 and 8.0, and a HO DVR centred around r0 = 1.821 with the mass and frequencycorresponding to the Morse potential. In both cases the grid size is varied until satisfactoryagreement with the exact eigenvalues is achieved.
4.2.3 State selective excitation
Having found a reliable representation of the model system, coupling to a light field can bestudied. Interaction with a time-dependent field E(t) is added to the Hamiltonian, resulting in
H = − 1
2m
∂2
∂r2+ V (r)− µ(r) · E(r) (4.5)
with the Morse potential V (r). The dipole moment is non-trivial in this system, as it dependson the internuclear distance. It is modelled by the Mecke function introduced in equation (2.21)with the parameters R∗ = 0.6 A= 1.134, µ0 = 7.85D/A= 1.634 [15].
Population analysis
Before turning to population control, a few words on how the populations of the desired eigen-states are measured. Unfortunately MCTDH offers no tool to directly do this, so populationanalysis is performed with a self-written Python script. MCTDH does give (when asked) theprimitive basis population, i.e. the contribution of every single basis function to the wavefunc-
43
tion. Given a fixed basis χj of size N , the basis population of each eigenstate φi was collected:
φi =N∑j=1
uijχj (4.6)
Once the matrix U = (uij) is stored, the eigenstate population of any wavefunction Ψ can befound by means of a simple basis transformation. Again, Ψ is first expanded in the primitivebasis and the coefficients are collected:
Ψ =N∑j=1
ajχj (4.7)
Ψ =21∑i=0
biφi (4.8)
with bi = 〈φi|Ψ〉 =N∑j=1
u∗ijaj (4.9)
The squared coefficients |bi|2 give the population pi of the eigenstate φi. Here it is assumedthat exactly 22 bound states exist, but of course the above formula is fully generalizable. Thiscalculation is exact and very fast, but it has the major drawback of working only for one specificcombination of primitive basis and eigenstates. As soon as the Hamiltonian or the DVR change,U has to be recomputed. The script also becomes useless if Ψ has significant contributions fromcontinuum states, when sums have to be replaced with integrals or, even worse, a combinationof both.
Rabi oscillations
The excitation energy from the ground state to the first excited state is E1 − E0 = 0.0172 =
3784.2 cm−1. Due to the anharmonicity of the system, the excitation to the next state alreadyshows a considerable detuning with E2 −E1 = 0.0164 = 3604.7 cm−1. This is large enough tomake the |1〉 → |2〉 transition rate negligible when exciting the first state resonantly. As a resultthe Morse oscillator can effectively be treated as a two-level system for sufficiently long pulsesand small amplitudes4.
As the first few states are sufficiently localized (∆r = 0.225 in state |1〉), the Mecke functioncan be linearised as in equation (2.22), yielding a dipole gradient of µ′ = 0.299 and a transitionmatrix element µ01 = µ′〈0|r|1〉 = 0.025445.
4The approximation breaks down for very short (. 100 fs) pulses because those have a larger bandwidth. For highamplitudes (E0 & 0.1 ≈ 50MV/m) the field can no longer be treated perturbationally.
5 There is actually no need to linearise the dipole operator since the integral µ01 = 〈0|µ(r)|1〉 can easily beevaluated numerically. The way given in the text is appealing because the matrix elements of the positionoperator are known analytically [6], e.g. β〈0|r|1〉 =
√2(N−1)2N−1 with N =
√2mDβ − 1
2 .
44
This results in a Rabi frequency of Ω = µ01 · E0 and an oscillation period of
TRabi =2π
Ω= 5.9746 fs · E−1
0
with E0 given in atomic units (e.g. T ≈ 1000 fs for E0 = 0.006 ≈ 3MV/m).The system is excited with monochromatic laser light of constant amplitudeE0 = 3.07MV/m
and the resonant frequency ωL = E1 − E0 = 3784.2 cm−1.The oscillations under a slight detuning of the laser frequencyωL = E1−E0±∆ are examined,
too. The maximum population of |1〉 should then be given by Ω2
Ω2+∆2 .Subsequently, Gaussian pulses of varying length are used. Complete population transfer is
attempted with π−pulses as well as 50% population by using a π/2−pulse. The field thus takesthe form
E(t) =N · E0√2π(∆t)2
e− 1
2
(t−tp∆t
)2
· cos(ωLt) (4.10)
where typical values for the centre and width of the pulse are ∆t ∼ 100 fs and tp ∼ 4∆t,respectively. The amplitude E0 can be left unchanged (compared to the constant amplitudesimulations) as long as the normalization constant N is chosen such that the area theorem isfulfilled, i.e.
∫Ω(t)dt = π ⇐⇒ N = TRabi/2 for a π-pulse.
This simulation is repeated with a sequence of two Gaussian pulses, the second one havinga laser frequency of ωL = 3604.7 cm−1 to populate the second excited state. The respectiveamplitudes areN1E0 = 123.5 (unchanged) andN2E0 = 0.699N1E0 due to the higher transitiondipole moment µ21 = µ10/0.699.
|0〉 to |5〉 to |10〉 excitation
Once the most basic example of population transfer has been mastered, interest returns to theresults in [15]. In this reference, the model system is excited from the ground state to the fifthexcited state by multi-photon absorption in a monochromatic sinusoidal pulse. The electric fieldis given by
E(t) = E1 · sin2
(πt
T
)· cos(ω1t) (4.11)
with the amplitude E1 = 328.05MV/m, the laser frequency ω1 = 3424.2 cm−1 and the pulselength T = 1 ps. This transfers the system into the |5〉 state.
After 1 ps another pulse with the same shape sets in to excite the bond further to the |10〉 state.Its parameters are E2 = 214.56MV/m, ω2 = 2524.7 cm−1 and identical pulse length of 1 ps.Note that longer wavelengths are required for high quantum numbers because the neighbouringstates lie closer.
The authors of [15] report a selective excitation of close to 100% with these pulses. In this
45
Figure 4.4: Laser pulses for excitation of the Morse oscillator to |15〉, adopted from [15]. Thesin2 shape of the pulses is clearly visible.
thesis, their pulse parameters are used in an attempt to achieve the same.
|10〉 to |15〉 excitation
Another multi-photon absorption process can drive the system from |10〉 to the target state |15〉.A final 1 ps-pulse with the parameters E3 = 148.07MV/m and ω3 = 1625.7 cm−1 is used forthis purpose. The full pulse train is depicted in figure 4.4.
Compared to the previous cases there is a number of differences: With an energy of E15 =
−0.482 eV the target state lies close to the dissociation threshold and couples significantly tothe continuum states. Therefore a complete occupation is not achieved in [15], but instead adissociation probability of 6.9% is reported. However the state selectivity, defined as the ratioof population in |15〉 and the population of all bound states combined, is as high as 99.23%.Another important factor is the spatial spread of the wavefunction. It turns out that a measurablepopulation of the grid border occurs, which must be handled with a CAP as described furtherbelow.
Finally, the authors of [15] reduced the length of the pulse train. For this purpose, the firstand the second pulses were shortened to 500 fs each. In addition, the pulses were overlappedwith the second pulse setting in at t = 350 fs and the third pulses at t = 600 fs. The frequenciesand amplitudes were adjusted to: ω1 = 3425.8 cm−1, ω2 = 2425.7 cm−1, ω3 = 8148.6 cm−1,E1 = 400.75MV/m, E2 = 214.56MV/m, E3 = 426.47MV/m.
The resulting field is depicted in figure 4.5. Note that ω1 and ω2 are almost unchanged while
46
Figure 4.5: Shortened pulse train for excitation of the Morse oscillator to |15〉, again adoptedfrom [15]. The individual pulse envelopes are indicated.
ω3 is much higher than before. The motivation behind this is to accomplish the excitation from|10〉 to |15〉 with a single overtone transition instead of a 5-photon-absorption like before. Thisreduces the coupling to the continuum states and hence the dissociation probability to about4.1%.
When the wavefunction hits the border of the grid, an non-physical reflection of the wavefunc-tion at the last grid point occurs, dramatically affecting the accuracy of the calculations. This isa common problem when working with dissociative modes on a finite grid, and the techniqueof Complex Absorbing Potentials (CAPs) is an established workaround [20]. Near the border ofthe grid, a negative imaginary potential term
VCAP(r) = −iη · (r − rc)2 ·Θ(r − rc) (4.12)
is added to the Hamiltonian. Here Θ(r) is the Heaviside step function, so the potential is onlynon-zero to the right of a critical value rc. The parameter η controls the strength of the potential,and other exponents than 2.0 may be used on the second factor.
This potential absorbs a wavefunction propagating beyond xc. Now the norm is no longerconserved, but this is the lesser evil compared to a reflected wavepacket. After all, the dissociatedpart of the wavefunction is rather small6 and only contributes to the continuum states, which shallnot be studied here.
6Anticipating the next chapter, it turns out that the final norm for the calculations performed here is 97.9%, so alittle over 2% of the population is lost.
47
The one-photon excitation is calculated first without a CAP, then with CAP using the param-eters η = 0.25, rc = 6.0. This strength parameter is quite typical according to the examples inthe MCTDH handbook [25], and the value of rc was chosen with the grid populations in viewand to allow a moderately strong CAP at the grid end point (VCAP(8.0) = −1.0 · i).
For each type of calculation, the total population of |15〉 and the state selectivity are computedand compared with the reference.
4.3 Optimal Control
Having reproduced previous results with known pulse parameters, attention turns to optimalcontrol with CRAB. The Morse oscillator from section 4.2 remains the relevant system, but thecontrol target is set in a slightly different manner.
4.3.1 One- and two-parameter control
As a preliminary proof of concept, optimization is performed with the amplitude E0 as theonly free parameter. The target is maximal population transfer from the ground state to |1〉 viaresonant absorption. The field is given by the pulse
E(t) = E0 · sin2
(πt
T
)cos(ωLt) (4.13)
with the fixed frequency ωL = 0.01724 = 3784.2 cm−1 and length T = 1000 fs. Since theenergy content of the field is proportional to E2
0 , a simplified penalty function is chosen. Thefull cost functional reads
J = 1− 〈|Ψ(T )|1〉|2 + αE20 (4.14)
with penalty factors chosen in the range from 0 to a 250. Now the functional is considered afunction J = f(E0) of one variable. A minimum of f with respect to E0 is searched for variousstarting values of E0.
The same process is repeated for variable ωL while keeping E0 fixed. Eventually, an opti-mization for both parameters simultaneously is performed.
The Python library scipy.optimize [14] contains a number of different optimization algo-rithms, most of which are tested. This includes the Nelder-Mead (NMS) method described insection 3.3, but also the modified Powell’s method, the conjugate gradient (CG) method and theBroydon-Fletcher-Goldfarb-Shanno (BFGS) algorithm7. The latter two require knowledge ofthe gradient of f , which they estimate numerically by comparing the function values at two veryclose points.
7 For all subsequent calculations either NMS or Subplex are used, therefore no explanation of the other algorithmsis deemed necessary here. For details see the SciPy documentation [14] and the references therein.
48
4.3.2 CRAB with ten parameters
Next a multivariate cost function is optimized using crab.To avoid the complications near the dissociation threshold, |5〉 is declared the target state
(starting from the ground state again), and the control time is set to T = 1000 fs. This is a verymodest goal compared to the above results, but after all this is CRAB’s first real benchmark incombination with MCTDH, and it seems sensible to give the algorithm a goal which is knownto be doable.
The electric field is expanded in a Fourier basis as in equation (3.29) with Nc = 5, so a totalof 10 parameters is optimized. The initial guess is given by a sin2 shaped pulse with ωguess =
0.0156 = 3424 cm−1, Eguess0 = 0.036 = 18.4MV/m. The photon energy is exactly 1
5(E5 −E0),
so this frequency can lead to multi-photon excitation just as in section 4.2.3. The amplitudeis lower here, so this pulse yields a target population of about 2%. As an envelope function,Λ(t) = sin2
(πtT
)is used. The penalty term is α
∫E(t)2dt with α = 10−5. This small penalty
factor helps avoid large field amplitudes while still making sure the cost functional is dominatedby the target population.
The choice of the frequencies ωk is paramount. A natural choice would be to use the transitionenergies ωeigen
k = Ek −Ek−1. Another reasonable choice are the harmonic frequencies ωharmk =
2πTk. At the heart of CRAB lies the idea not to use the exact frequencies, but to randomize them
by multiplication with a random factor:
ωk = ωk ·(1
2+ rk
)(4.15)
with rk taken from a uniform random distribution on the the interval [0, 1] [5]. To avoid skewedresults by a particularly (un)fortunate choice of random numbers, this process is performed threetimes on the resonant and the harmonic frequencies. Finally, two sets of completely randomfrequencies are generated from a uniform distribution on [0, 0.2] (since E0 ≈ −0.2). All thefrequencies tested are summarized in table 4.1.
The results of section 5.3.2 will suggest that NMS is the most suitable algorithm withinscipy.optimize. As this library does not contain an implementation of Subplex, the C li-brary NLopt [13] is used. Amongst others, NLopt implements Subplex and also NMS, givingan opportunity to compare the two implementations of NMS. All three algorithms (NMS fromscipy, NMS from NLopt and Subplex from NLopt) are tested with each set of frequencies. Thecalculations are limited to 200 function calls each, with each call (MCTDH run) taking about 3seconds on a Linux server with about 3 GHz8
The penalty function is α∫ T
0E(t)2dt, but with a fairly small penalty factor of α = 10−5.
The cost function is dominated by the target state population p5 = |〈Ψ(t)|5〉|2, which is also thequantity of interest in the analysis.
8Most calculations were performed on DESY’s PAL cluster. The PAL servers have 2.4 to 3.0 GHz CPUs; the latterwas used in most cases.
49
Description Abbr. ω1 ω2 ω3 ω4 ω5
Exact resonant frequencies eig 0.017242 0.016425 0.015607 0.014789 0.013971Randomized res. freq. 1 eig-rn1 0.015714 0.017487 0.021671 0.017012 0.019498Randomized res. freq. 2 eig-rn2 0.025242 0.018605 0.021464 0.012774 0.020827Randomized res. freq. 3 eig-rn3 0.025472 0.014188 0.021326 0.014171 0.085043
Exact harmonic frequencies harm 0.000152 0.000304 0.000456 0.000608 0.000760Randomized harm. freq. 1 harm-rn1 0.000104 0.000203 0.000454 0.000767 0.000563Randomized harm. freq. 2 harm-rn2 0.000181 0.000430 0.000509 0.000736 0.000563Randomized harm. freq. 3 harm-rn3 0.000089 0.000217 0.000294 0.000694 0.000543
Random frequencies 1 rand1 0.199047 0.083381 0.038444 0.002658 0.051437Random frequencies 2 rand2 0.092450 0.186849 0.094596 0.168496 0.110986
Table 4.1: The frequencies used for ten-parameter control. The second column introduces ashorthand which will later be used.
4.3.3 Frequency optimization
Based on the advice of Dr. Antonio Negretti, the above algorithm is modified to also optimize thefrequencies. To keep the number of parameters approximately constant, the basis size is reducedto Nc = 3, turning the cost functional into a function of nine variables (ak, bk, ωk), i = 1, 2, 3.The starting values for ωk are the first thee harmonic frequencies from table 4.1, while the initialfield guess is again a sinusoidal pulse which results in a target population of 2%.
The Subplex and the NMS method (both from NLopt) are compared. Hope is that modulationwith the optimized frequencies yields better pulses than the fixed frequencies.
4.3.4 LiCs diatomic
The LiCs diatomic is an heteronuclear alkali molecule which has recently garnered interest boththeoretically and experimentally, mostly in the field of low temperature physics [2, 24]. It isformed by non-elastic collisions in a mixture of ultracold (< 1mK) Li and Cs vapour, but canalso be synthesized in a heat pipe at temperatures of a few hundred C for spectroscopic studies.This was done by P. Staanum et al. [24], who determined the ro-vibrational spectrum of the LiCsground state. The potential energy curve shown in figure 4.6, together with the electric dipolemoment µ(R) calculated by M. Aymar and O. Dulieu with ab initio methods [2]. Like similaralkali dimers LiCs has a large dipole moment.
The potential energy function has some resemblance to a Morse potential, as had already beenillustrated in figure 2.2. The major difference lies in the long-range behaviour, where the LiCspotential approaches zero proportional to −1/R6 (just like most diatomics [8]). The molecule’sdissociation energy has been determined experimentally to be 0.728 eV, while theoretical predic-tions suggest 0.743 eV. The experimentally found potential supports 55 bound vibrational statesfor a non-rotating molecule [24].
This system is modelled on a sine DVR with basis size 96 with the inter-nuclear distance
50
Figure 4.6: Potential energy surface and electric dipole moment of the LiCs diatomic. Datafrom [24, 2], kindly provided by Dr. Peter Schmelcher.
ranging from 5.0 to 20.0. At x = 18.0, a CAP with the strength parameter η = 0.25 sets in toavoid possible reflections of the wavepacket with the grid edge.
In order to probe state selective control with short pulses, the control time is set to T = 250 fs.The target state is |1〉 with an energy difference of E1 − E0 = 22.7meV= 183 cm−1. Theexcitation of the target state is attempted via resonant one-photon absorption, hence an initialpulse with ωL = 183 cm−1 is used. The pulse envelope is again a sin2 function with an amplitudeof E0 = 5 · 10−4, yielding a target population of 37.5%.
For optimization of the guess field, the frequency optimization scheme is employed again,using the harmonic frequencies ωk =
2πTk = 6.08 · 10−4 · k for k = 1, 2, 3.
The results are compared with ten-parameter CRAB. The fixed basis set uses the randomizedharmonic frequencies ωk = 0.68 · 10−3, 1.67 · 10−3, 1.73 · 10−3, 2.40 · 10−3, 4.47 · 10−3.
In both cases the NMS and the Subplex (both from NLopt) method are used, again allowinga maximum of 200 function calls.
4.3.5 Krotov scheme
Results of the Morse and the LiCs system are compared with the field gained by MCTDH’soptimal control tool optcntrl using the Krotov optimization scheme. It requires no frequencybasis since it searches the entire space of numerically integrable functions. The physical inputparameters as well as the initial guess pulse are identical, but the cost functional is given by
J = 1− ptarget + α
∫ T
0
E(t)2
sin2(πtT
) dt (4.16)
51
The difference lies in the penalty term, where the sin2 ensures a smooth pulse. The penalty factorα is chosen to 0.01 for the Morse oscillator and 0.1 for the LiCs system (where overall smallerfield amplitudes are used). The algorithm is execucted for 10 iterations, which involves a totalof 20 propagations (forward and backward in time) and takes about 30 minutes.
Given the larger search space and the computational cost of the Krotov scheme, it is expectedto give the best result of all compared methods.
52
5 Results and Discussion
This chapter comprises the results of all calculations described in chapter 4 while following thesame structure.
5.1 Harmonic Oscillator
5.1.1 Propagation
The wavefunction propagation worked as expected. The probability densities |Ψ(x, t)|2 for dif-ferent initial wavefunctions are displayed in figure 5.1. When the initial wavefunction was aneigenstate, no change over time was observed. The coherent states oscillated around zero withthe harmonic frequency ω while the shape of the wavepacket did not change.
Every non-stationary state oscillates with this frequency. This can be shown analytically byrepresenting the initial wavefunction in terms of the HO basis and propagating the expansionexplicitly:
Ψ(0) =∞∑n=0
cnφn (5.1)
Ψ(t) =∞∑n=0
cne−iEntφn = e−
iωt2
∞∑n=0
cne−i·nωtφn (5.2)
using the known harmonic energiesEn = ω(n+ 12). Apart from the irrelevant total phase e− iωt
2 ,the wavefunction is periodic with the frequency ω.
Finally, the squeezed coherent state increased and decreased in width periodically, this timewith the frequency 2ω. Al tough it looks Gaussian upon first glance, the wavefunction can beshown to vary in a more sophisticated manner. The coefficients cn from equation (5.1) can bederived explicitly by calculating the scalar product of the initial Gaussian (which correspondsto the ground state of a HO with the same mass, but frequency Ω). Working in units where
53
Figure 5.1: Probability densities in the HO. Top left: Ground state. Bottom left: First excitedeigenstate. Right: Coherent state, initially centred around x = 10. All times in fs.
mω = 1, one finds:
cn = 〈Ψ(x, t = 0) | φn(x)〉 (5.3)
=
∫ ∞
−∞
[(Ω
π
) 14
e−Ω2x2
]·
[1√2nn!
(1
π
) 14
e−12x2
Hn(x)
]dx (5.4)
=1√2nn!
Ω14
√π
∫ ∞
−∞e−
12(Ω+1)x2
Hn(x) dx (5.5)
The first two Hermite polynomials are H0(x) = 1 and H1(x) = 2x. Therefore c1 vanishes bysymmetry (integral of an odd function), while c0 reduces to the standard result
c0 =Ω
14
√π
∫ ∞
−∞e−
12(Ω+1)x2
=Ω
14
√π·√
2π
Ω + 1= Ω
14
√2
Ω + 1(5.6)
It is interesting to take a look at the special case Ω = 1. Since H0 = 1, equation (5.5) can be
54
interpreted as the scalar product of H0 with Hn with the weighting function e−x2:
Ω = 1 ⇒ cn =1√
2nn!π
∫ ∞
−∞e−x2
Hn(x) dx (5.7)
=1√
2nn!π
∫ ∞
−∞H0(x)e
−x2
Hn(x) dx (5.8)
=1√
2nn!π〈H0|Hn〉 (5.9)
= δ0n (5.10)
With respect to this scalar product the Hermite polynomials are orthogonal. Thus when theinitial wavefunction has the correct width, it just coincides with the ground state.
For arbitrary Ω, a recursive relation for cn can be found via integration by parts while exploit-ing the fact that the Hermite polynomials satisfyHn = 2xHn−1−2(n−1)Hn−2. The calculationis straightforward, but the exact values of cn are not very important (cn+2 turns out to be simplyproportionial to cn). A look at the integral immediately reveals that c1 = c3 = . . . = 0, as allodd-numbered Hermitian polynomials are, in fact, odd functions. Hence only the even-indexedcoefficients c0, c2, . . . contribute to Ψ(t), adding terms which are periodic in 2ω, 4ω, . . .. Thisexplains why the period of the observed oscillation is 5 fs rather than 10 fs. In figure 5.2 theevolution of the density is plotted. For better spatial resolution, a grid size of 48 (instead of 12a usual) was used to produce these figures.
5.1.2 Relaxation
It turned out that a modest number of basis functions is sufficient, but the default integratorsettings need some adjustment for accurate relaxation of a wavefunction. Interestingly, bad set-tings result in a final energy below the ground state energy E0. The algorithm converges rapidly(within less than one imaginary fs) to the correct energy scale, but then slowly falls further inenergy.
The default integrator for the SPFs is Bulirsch-Stoer (BS) with an accuracy of 10−6. It deviatesfrom the exact result E = 217.691071 eV by about 0.365 eV, which is too large an error forquantum chemical calculations. By setting the integrator accuracy up to 10−8, the result couldbe improved to acceptable accuracy, but it was still worse than other integrators.
Better results were obtained with the Adams-Bashforth-Moulton (ABM) integrator and theRunge-Kutta (RK) integrators (order 5 and order 8 were available and showed comparable per-formance). Both gave much better results than BS at an accuracy of 10−6 and were exact in allnine significant digits when using an integrator accuracy of 10−8.
A larger primitive basis did not improve the result, and even 12 basis functions proved suf-ficient for exactness. No significant difference was observed between the VMF and the defaultCMF (variable/constant mean field) integration scheme. Some of the tested configurations are
55
Figure 5.2: Density of a squeezed coherent state, evolving in time. Bottom left: Densities atdifferent times. The 4 fs density coincides with the 1 fs density, same for 0 fs and5 fs. Bottom right: Density at t = 1.25 fs, compared with the ground state density.Note that |Ψ(x)|2 is smaller than the Gaussian at the centre, but higher at the flanks.Top: 3d view of the density.
56
Figure 5.3: Accuracy of the relaxation with different integrators, using the CMF scheme. Thebars represent the relative error of the calculation |1−Ecomputed/Eexact|. The numberafter the integrator name (see text for abbreviations) is the integrator accuracy.
compared in figure 5.3.
5.1.3 Coupling to a field
An interaction such as (4.3) effectively shifts the zero of the potential to
x0 = µ01
2E(t) (5.11)
without changing the harmonic features of the oscillator. This system has been solved explicitlyin the framework of Floquet theory [9]. The fundamental solutions of the TDSE are the so-calledFloquet modes
Ψn(x, t) = ϕn(x− ξ(t))eiφ(t) (5.12)
where ϕn are the eigenfunctions of the unperturbed HO, the coordinate shift ξ(t) obeys theequation of motion
ξ(t) + ω2ξ(t) = µ0E(t)/m (5.13)
and the phase φ(t) is a functional depending on time and on ξ(t) (see the literature [9] for theexplicit form). As a result, this type of interaction just creates a coherent state, but is incapable
57
Figure 5.4: Wavepacket in a monochromatic field, off-resonant with ωL = 2ω (left) and resonantωL = ω (right). The artefacts after 12 fs in the resonant case stem from reflection ofthe wavepacket at the grid borders.
of selectively exciting specific eigenstates. This can be checked by inspecting the occupationnumbers n given in the output file of MCTDH: At any time ∆n =
√〈n〉 holds, just as is known
for coherent states.A monochromatic field E(t) = E0 cos(ωLt) leads to a classical forced oscillation. Solving
the equation of motion for ξ(t) yields [9]
ξ(t) =µ0E0
m
1
ω2L − ω2
cos(ωLt) (5.14)
The amplitude shows the typical Lorentzian dependency on the frequency, including divergencefor the resonant case ωL → ω. Both the off-resonant and the resonant behaviour could bereproduced1 with MCTDH as can be seen in figure 5.4. Finite-length pulses excite the pulseto a coherent state. After the pulse ends, the system oscillates just as in section 5.1.1. Onerepresentative example is depicted in figure 5.5.
5.2 Morse Oscillator
5.2.1 Choice of DVR
As mentioned before (section 4.2), eventually a primitive basis of 96 sine functions was chosenspanning a grid from x0 = 0.7 to x97 = 8.0. This is justified by the results of the followingparagraph. Until this insight was gained, a HO DVR withN = 64 had been used for propagation.This should not affect the accuracy of those calculations, although the calculation time wasunnecessarily long.
1In case of resonance the wavefunction soon hit the borders of the grid, which is interpreted as a rise to infinity.
58
Figure 5.5: Light pulse (top left) exciting the wavepacket in the HO potential. All pulses depictedin figure 4.3 resulted in a qualitatively similar behaviour.
5.2.2 Diagonalization
All 22 eigenenergies of the Morse oscillator could be calculated with MCTDH with a primitivebasis of size 128. The calculations in the HO basis accurately returned the first 10 to 15 energies,but failed dramatically for the higher states. The sine DVR from x = 0.7 to 8.0 gave a muchbetter overall performance, as can be seen in figure 5.6. Only the last bound state was determinedincorrectly when MCTDH insisted that E21 > 0.
This could be improved by stretching the grid to an upper limit of x = 12.0 while keeping thenumber of basis function at 128. With this DVR all eigenenergies including E21 were found togreat accuracy, see table 5.1 for the results.
It is concluded that a sine DVR is the best choice for possibly dissociative modes. Since nosimulations directly involve states higher than |15〉, the smaller grid (upper limit 8.0) was chosento achieve higher spatial resolution at moderate internuclear distance.
5.2.3 State selective excitation
The coupling to a light field described in section 4.2.3 gave the expected results. The first para-graph of this section presents the Rabi oscillations, while the following paragraphs describe thepath to selective excitation of the |15〉 state.
Rabi Oscillations
In a monochromatic light field E0 cos(ωLt), the Morse oscillator performed Rabi oscillationswith the predicted amplitude and Rabi frequency Ω. Figure 5.7 shows the evolution the popula-
59
n En [eV] MCTDH error0 -5.18301 +1.06 · 10−12
1 -4.71383 +3.77 · 10−12
2 -4.26690 +5.05 · 10−12
3 -3.84222 +1.23 · 10−11
4 -3.43981 +1.01 · 10−10
5 -3.05964 +3.56 · 10−10
6 -2.70174 −5.00 · 10−10
7 -2.36608 −7.99 · 10−9
8 -2.05269 −2.51 · 10−8
9 -1.76154 −1.53 · 10−8
10 -1.49266 +1.47 · 10−7
11 -1.24603 +6.28 · 10−7
12 -1.02165 +1.39 · 10−6
13 -0.81953 +1.87 · 10−6
14 -0.63967 +8.37 · 10−7
15 -0.48206 −3.25 · 10−6
16 -0.34671 −1.17 · 10−5
17 -0.23361 −2.56 · 10−5
18 -0.14276 −4.64 · 10−5
19 -0.07418 −7.90 · 10−5
20 -0.02784 −1.44 · 10−4
21 -0.00377 −7.39 · 10−3
Table 5.1: Eigenenergies of the Morse oscillator as determined from equation (2.14). The lastcolumn gives the relative error of the diagonalization with MCTDH when using the128-point sine DVR ranging from 0.7 to 12.0.
Figure 5.6: Selected eigenenergies found by diagonalization of the Morse Hamiltonian as a func-tion of basis size. Left: HO DVR. Right: Sine DVR. The dashed lines represent theexact energies.
60
Figure 5.7: Rabi oscillations of the Morse system in a monochromatic field with ωL =3784.2 cm−1 and E0 = 5.9746 · 10−3 = 3.07MV/m. According to (4.2.3) thiscorresponds to a Rabi period of 1 ps. The points are the time-dependent populationsp0 and p1 along with the fitted cos2(Ωt) resp. sin2(Ωt) functions. The curve at thebottom illustrates the quality of the two-state approximation, as the population p2reaches 0.02 at most. A closer look reveals that the Rabi period of this oscillation isnot exactly 1 ps, but rather 992 fs. It turns out that the exact resonance frequency isslightly higher, see the text and figure 5.9.
tions pn = |〈Ψ(t)|n〉|2 for one and a half Rabi cycles in the resonant case. An off-resonant fieldwith the frequency ωL = ω+∆ led to the expected decline in population transfer and increasedRabi frequency Ω′ as given by
Ω′ =√Ω2 +∆2
max(p1) =Ω2
Ω2 +∆2
(5.15)
One example of a detuned Rabi oscillation is depicted in figure 5.8 while figure 5.9 shows datacollected from a number of such simulations with different detuning. Equations (5.15) are shownto hold, and the fits show the resonant frequency to lie at 3787.4 cm−1. This is about one percentlarger than what would be expected from the difference of eigenenergiesE1−E0 = 3784.2 cm−1.
61
Figure 5.8: Rabi oscillations in a detuned laser field with the frequency ωL = 0.991ω.
After checking the accuracy in the input parameters, the most likely explanation appears to be asystematic error such as coupling to the |2〉 state.
Population transfer was also performed successfully with Gaussian pulses with the frequencyωL = 3784.2 cm−1. Figure 5.10 illustrates the transfer of the system into state |1〉 and thecoherent superposition |0〉+|1〉√
2with a π-pulse and a π/2-pulse, respectively. After successful
excitation with one π-pulse, a sequence of two Gaussian pulses was used to excite the systeminto |2〉 via |1〉. The first pulse again had a frequency of ω10 = 3784.2 cm−1, while the secondpulse was tuned to excite the next transition at ω21 = 3604.7 cm−1. The population over time isplotted in figure 5.11.
Finally, there were also cases of failed selective excitation. One example of a failed populationtransfer to |1〉 is depicted in figure 5.12. In this case, an attempt was made to complete thepopulation transfer in just 50 fs, i.e. just a few optical cycles.
The derivation of the Rabi oscillations relies on the Rotating Wave Approximation (RWA)which states that the coefficients cn = 〈Ψ(t)|n〉 change on a much longer timescale than theexternal field [11]. This is no longer the case here, so the RWA breaks down. Also, a pulse2 witha width of 25 fs in the time domain has a spectral width of over 200 cm−1. With the excitationenergy ω21 lying closer to ω10 than this, significant excitation from |1〉 to |2〉 occurred.
2To be clear, a constant amplitude field was used. But on this time scale the difference between such a field and alight pulse blurs.
62
Figure 5.9: Detuned Rabi oscillations. Amplitude (top) and Rabi frequency (bottom) as a func-tion of the laser frequency, along with fits of the form (5.15). The extrema lie at aresonant frequency of 3787.4 cm−1.
63
Figure 5.10: Top: A π-pulse with the parameters E0 = 5.9746 · 10−3 and the normalizationconstant (see equation (4.10)) N = T/2 = 2.07 · 104 excites the system into state|1〉. The pulse is centered around t = 800 fs with a width of 200 fs (see inset). Thetarget population reached a final value of p1 = 99.3%. Bottom: π/2-pulse with thesame parameters as above except for the amplitude, which was halved.
64
Figure 5.11: Two Gaussian pulses centered around 0.5 ps and 1.1 ps with a width of 0.2 ps eachexcite the system into |2〉. The final population was p2 = 99.1%.
Figure 5.12: Failed population transfer due to a very short, intense pulse.
65
|0〉 to |15〉 to |10〉 excitation
The pulse parameters described in [15] and reprinted in section 4.2.3 were used to successfullytransfer the system to |5〉 and then to |10〉 via multi-photon absorption with two sin2-shapedpulses of 1 ps length. The target populations reached a value of p5 = 99.996% after 1 ps andp10 = 99.986% after 2 ps. Unfortunately these values cannot be compared with the originalpaper, which is as unspecific as “close to 100%” at this point.
Notably, the excitation to |10〉 was the point when the HO DVR failed. While the harmonicoscillator with size N = 64 basis was good enough to model the system up to |5〉, an accuratesimulation of the next step required a more suitable DVR. After changing to a sine DVR of thesame basis size, the results matched those reported in [15]. See figure 5.13 for a comparison. Inaddition the CPU time required for the calculations decreased by a factor of 5 to 10.
|10〉 to |15〉 excitation
The new grid ranged from x = 0.7 to 6.0, which was apparently sufficient for an accurate cal-culation up to state |10〉. However, this DVR failed as well when further excitation to |15〉 wassimulated. Hence it was extended to x = 8.0 and a basis size of N = 96. The differenceis clearly visible in figure 5.14. Here “Well” denotes the cumulative population of all boundstates, i.e. pWell =
∑21n=0 pn. Hence the dissociation probability is D = 1 − pWell, which
amounts to 4.15% when using the larger grid. The target population amounted to p15 = 94.8%,so the state-selectivity defined as S = p15/pWell is 98.5%. This last value is only in approxi-mate agreement with [15], where a value of S = 99.2% was reported. The target populationand dissociation probability are even less accurate, as the reference values are p15 = 92.4 andD = 6.9%. The simulations performed in this thesis seem to drastically underestimate the dis-sociation probability. This point will be addressed further below.
However the results were in qualitative agreement with results of the reference which areshown in figure 5.15 (top). Therefore no modifications to the input were when turning to the lastexcitation scheme presented in [15]. Here the final excitation step was simulated as a one-photonprocess by setting the frequency of the final pulse to the overtoneω3 = 8148.6 cm−1. At the sametime the pulse train was compressed to a total length of 1600 fs as described in section 4.2.3.
The result of the calculation can be seen in figure 5.16 (top) and should be contrasted with theknown result shown in figure 5.15 (bottom). Here the failure to predict the dissociation probabil-ity was even more evident. The authors of [15] reported a dissociation probability ofD = 4.14%,a target state population of p15 = 94.54% and a state selectivity of S = 98.62%. The resultsobtained here were D = 0.24%, p15 = 98.45% and S = 98.69%, respectively. Although thestate selectivity happened to agree with the literature, the other values are considerably off.
Checking the population of the last few grid points revealed that part of the wavefunction hadindeed propagated all the way to x = 8.0 where it was reflected at the edge of the grid. In sucha case a CAP as described in section 4.2.3 is called for. The simulations were repeated with
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Figure 5.13: Excitation of the Morse oscillator to the target state |10〉 with two monochromaticpulses. The plots show the populations of the most important states over time andonly differ in the DVR used. Top: HO DVR of size N = 64. Bottom: Sine DVRof the same size.
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Figure 5.14: Excitation to the target state |15〉 with a sequence of three pulses (see figure 4.4).Both plots are the result of calculations with a sine DVR. Top: Basis size 64, x =0.7 . . . 6.0. Bottom: Basis size 96, x = 0.7 . . . 8.0.
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Figure 5.15: Population dynamics taken from [15]. Top: Population of |15〉 via multi-photon ab-sorption as in figure 5.14 (bottom). Bottom: Shortened pulse train with an overtonetransition to excite |15〉. Compare to figure 5.16 (bottom).
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Figure 5.16: State populations with an overtone transition from |10〉 to |15〉 without (top) andwith (bottom) CAP.
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the CAP parameters described there, leading to the populations shown in figure 5.16 (bottom).The improvement is striking and also the numerical results are much more convincing withD = 4.23%, p15 = 94.36% and S = 98.53%. The final wavefunction had a norm of
∫|Ψ|2dx =
0.9786, so about 2.1% of the total population had absorbed away by the CAP. The norm remainedlarger than 0.999 until about 700 fs, when a significant part of the wavefunction approached thegrid edge.
The dissociation probability was thus determined to be slightly larger than expected. Thismight be due to the CAP damping some part of the wavefunction which was just below thedissociation threshold, but still in a high bound state. A possible remedy would be to use aneven larger grid (e.g. ranging to x = 12.0) and shifting the CAP further to the right. This wouldmake sure that only the truly dissociative parts of Ψ hit the CAP. However such refinements areleft to future studies.
As a conclusion of this subsection, the importance of a suitable DVR and CAPs has beenhighlighted. As soon as higher excited states are involved, a sine DVR or a similar primitivebasis is inevitable for reliable results. A HO DVR might still work accurately at medium quantumnumbers (near |5〉), although the sine DVR is still much more efficient there. When setting theborders of the grid, care has to be taken such that all involved states (in the dissociative case,this usually includes all bound states) can be represented on the grid. Unless the grid is muchlarger than necessary, part of the wavepacket is still likely to hit its edge. Albeit unphysical, aCAP can significantly improve the results of the simulation in this case.
5.3 Optimal Control
With the DVR and the CAP set up as described, various optimal control calculations were per-formed. The optimized field would usually not perform as impressively as the examples above,but in many cases they could improve significantly over the initial guess pulse Eguess(t).
5.3.1 One- and two-parameter control
For a resonant pulse with ωL = 0.01724 = 3784.2 cm−1 the optimal3 amplitude was foundto be Eopt
0 = 0.0059 = 3.03MV/m, yielding a target population of p1 = 99.3%. Likewisethe frequency was optimized while keeping the amplitude fixed, but the result ωopt
L = 0.01725
deviated just very slightly from the resonant frequency. Figure 5.17 shows a full scan of theregions of interest in both cases. The multiple peaks in p1(E0) can be explained by interpretingthe population transfer as a Rabi oscillation with a Rabi frequency proportional to E0. Fullpopulation of |1〉 occurs whenever
∫Ω(t)dt = π, 3π, 5π, . . ., and an increasing amplitude will
periodically fulfil this condition at E0 = 0.018, 0.03 . . ..
3In the sense of maximizing target population, i.e. for a penalty factor α = 0.
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Figure 5.17: Target population p1 and penalty function J as a function of the pulse amplitude(top) or frequency (bottom) while keeping the other quantity fixed at ωL = 0.01724resp. E0 = 0.046 (non-optimal on purpose to leave room for improvement). Thepenalty function was J = 1 − p1 + 250E2
0 in the first and J = 1 − p1 + 50ωL inthe second case.
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Figure 5.18: Convergence path in the E0-ωL-plane. Starting from a guess E0 = 0.004, ωL =0.017, the NMS algorithm found a minimum of the cost functional at (Eopt
0 , ωopt) =(0.00593, 0.0172) (all in au). The final population of |1〉 reached p1 = 99.98%.
Rather surprisingly, the gradient-based algorithms in scipy.optimize failed to minimize Jin both cases. While the simplex algorithm (NMS) would find at least a local minimum for moststarting values, all other algorithms consistently diverged to very large or even negative valuesof the variables within a few iterations. Apparently the numerical derivative of the cost func-tional is unreliable, which is somewhat surprising given the smooth appearance of the curvesin figure 5.17. This behaviour could not be alleviated with a different penalty factor or pre-conditioning of the problem (i.e. variable transformations to avoid numerical instabilities).
It is also not a consequence of the pathological nature of the one-dimensional case. Thetwo-dimensional optimization via amplitude and frequency was equally unsuccessful with allgradient methods, while NMS converged quite reliably, see e.g. figure 5.18. Even with NMS theguess for ωL must not be too far off, as a strongly detuned pulse will effectively return p1 = 0,so the algorithm has little to work with.
Therefore only direct search methods such as NMS were employed in the following calcula-tions. This is in accordance with the original paper on CRAB [5], which also promotes this typeof algorithms.
5.3.2 CRAB with ten parameters
The initial pulse guess and the resulting populations are depicted in figure 5.19.Attempts to optimize this pulse with a modulation of the type (3.29) gave mixed results de-
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Figure 5.19: Left: Initial guess field with Eguess0 = 0.0036, ωguess = 0.0156, T = 1000 fs. States
populations after t = T . The target state |5〉 is populated with 1.98%.
pending on the quintuple of used frequencies and the optimization algorithm. These findingsare discussed organized by the family of basis sets (resonant, harmonic or fully random frequen-cies), the application of CRAB (i.e. randomization of the basis) and the employed minimizationalgorithm (NMS from scipy, NMS from NLopt and Subplex from NLopt).
• On average the harmonic frequencies performed best with target populations consistentlyabove 90%. The purely random frequencies turned out to be not very useful. One set ofrandom frequencies failed completely in all optimization runs while the other convergedto target population of barely over 50%. The resonant frequencies of the system gaveoverall satisfactory results, often reaching populations over 90%. Although the singlebest optimization run was made with one frequency set from this family (eig-rn1), otherswould occasionally fail and usually performed worse than the harmonic frequencies.
• Randomization in the spirit of CRAB showed little benefit here. Although one set ofrandomized resonant frequencies gave a superior result, all other randomized basis setsperformed consistently worse than the exact frequencies. This applies both to the resonantand the harmonic families.
• An astonishing gap in performance was found between the scipy.optimize and theNLopt library. In all but one case, the NLopt algorithms converged much faster and tomuch better results than the scipy algorithm did. The reason is not entirely clear, as bothNMS implementations employ the same strategy, i.e. use the same parameters for simplexevolution. Perhaps scipy chooses the initial simplex in an unfortunate manner. Whencomparing the two algorithms in the NLopt library, Subplex and NMS fared comparablywell.
These results are summarized in table 5.2. Figures 5.20 to 5.22 illustrate the points made above.The fields resulting from the resonant frequencies were very complex in shape with peak
values near 0.2 au. The harmonic frequencies (exact or randomized) led to smoother pulses with
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Table 5.2: Final target populations for all combinations of basis sets and algorithms after 200function calls each. For details on the frequencies see table 4.1. A final value of0.020 indicates no improvement over the initial guess.
Frequency set scipy-NMS NLopt-NMS NLopt-Sbplxeig 0.972 0.872 0.806
eig-rn1 0.992 0.994 0.999eig-rn2 0.020 0.020 0.020eig-rn3 0.472 0.669 0.572harm 0.521 0.972 0.994
harm-rn1 0.819 0.961 0.976harm-rn2 0.748 0.972 0.975harm-rn3 0.646 0.991 0.940rand1 0.020 0.587 0.553rand2 0.020 0.020 0.020
Figure 5.20: Performance of the different optimization algorithms with the exact harmonic fre-quency basis (harm). The plot shows the infidelity 1 − p5 versus the number offunction evaluations. The Subplex method (Sbplx) worked best in this case with afinal infidelity of just 0.006.
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Figure 5.21: Same as figure 5.20, but with the randomized frequency sets harm-rn3 (top) andeig-rn1 (bottom). The respective lowest infidelities were 0.009 and 0.0014.
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Figure 5.22: Comparison of different frequency basis sets. All calculations executed with Sub-plex. Top: The eig family with the resonant frequency set and its randomizations.Bottom: The harmonic frequency set harm and randomized copies. Except foreig-rn1, which yielded a large improvement over eig, the randomized frequen-cies performed not better than the exact ones. Contrasting the two diagrams alsoreveals that the optimization converges more reliably with harmonic frequencies.
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Figure 5.23: Two fields optimized with the eig-rn1 basis by the NLopt NMS method (left) andSubplex (right).
Figure 5.24: Same as figure 5.23, but with the frequencies harm.
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amplitudes below 0.1 au. This is due to the fact that the harmonic frequencies are smaller by oneto two orders of magnitude, allowing only rather slow modulations of the pulse. Some typicalexamples are depicted in figure 5.23 and 5.24.
5.3.3 Frequency optimization
The optimization with variable frequencies yielded excellent results. Although the search spacewas smaller than in section 5.3.2 (nine- instead of ten-dimensional), the final infidelities were0.0062 for Subplex and 0.0014 for NMS, well below those of most fixed frequency runs.
The infidelities are depicted in figure 5.25. The Subplex method showed its efficiency by con-verging much faster within the first iterations, but did not improve much after that. Apparentlyit ignored some important subspaces which were slowly, but steadily explored by NMS. Thispicture might change for other initial values or a different optimization target.
The final frequencies found by NMS were ~ω = (0.90 · 10−4, 2.84 · 10−4, 4.52 · 10−4), consid-erably different from the initial values of (1.52 · 10−4, 3.04 · 10−4, 4.56 · 10−4). Still, the finalpulse (figure 5.26) bears a lot of similarity with the results of ten-parameter optimization withthe harm basis, as it is relatively smooth and has a moderate amplitude.
This result highlights the importance of the frequencies used for optimization. Unless one issure to have an ideal basis set, it is generally advisable to also optimize the frequencies for bestperformance. This goes against the spirit of CRAB, where a randomized, but fixed basis is used.However, randomized frequencies might still be a good initial guess for optimization.
5.3.4 LiCs diatomic
This system turned out to be very difficult to control. No way was found to populate exclusively|1〉 without simultaneously exciting other states.
Frequency optimization led to target populations of 42.5% with NMS and 43.8% with Sub-plex. CRAB, using the randomized harmonic basis, achieved comparable populations of 46.4%and 43.2%, respectively. In this case, the randomized basis performed better than the optimizedone, if just slightly. Figure 5.27 shows the convergence behaviour, and the best field found byCRAB is shown in figure 5.28.
The reason for these disappointing results lies in the energy structure of LiCs. At low energies,the system has negligible anharmonicity and almost equidistant eigenvalues. This can also beseen in table 5.3, which lists the first 20 eigenenergies of LiCs found by diagonalization withMCTDH. As a result, small perturbations excite the system into an near-coherent state, whichalways has significant contributions from more than one eigenstate. Although more iterations ofCRAB might have improved the target population a little, it cannot possibly get close to 100%.
As a remedy, one might suggest excitation into higher states such as |10〉 or even |20〉. Un-fortunately, such calculations are impossible to perform accurately with the used primitive grid.Its lower border was as large as x = 5.0 – a point where the potential almost crosses zero. A
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Figure 5.25: Optimization with variable frequencies. Infidelity 1 − p5 over the number ofMCTDH runs.
Figure 5.26: Fields optimized with a variable frequency basis, using NMS (left) and Subplex(right).
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Figure 5.27: Optimization of the |0〉 to |1〉 transition in LiCs. CRAB with randomized harmonicfrequencies was compared to the variable frequencies scheme described in sec-tion 4.3.3. CRAB with the NMS algorithm showed the best result with a finalpopulation of 46.4%. Note that the ordinate axis is not logarithmic.
Figure 5.28: Optimized field for excitation of LiCs, found with CRAB and the NMS method.The pulse is depicted on the left, the final population on the right.
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Table 5.3: Lowest eigenvalues of the LiCs molecule, determined with the diagonalization toolof MCTDH. Note that the molecule has at least 55 bound states, but the calculation ofhigher eigenenergies is rather inaccurate. After some experimentation with differentDVRs and integrator settings, the error in E19 is presumed to be at most 1 meV andconsiderably smaller for the lower states.
n En [meV] n En [meV]0 -717 10 -5021 -694 11 -4822 -672 12 -4623 -650 13 -4434 -628 14 -4235 -606 15 -4046 -585 16 -3867 -564 17 -3688 -543 18 -3489 -522 19 -332
larger grid would be very desirable, but no data for the electric dipole moment were available forx < 5.0. No sensible method for extrapolation was evident, therefore all attempts to selectivelyexcite higher states were abandoned.
Given more time, a more detailed study of this system might have been rewarding. Moreeffort should be placed on finding the optimal DVR in order to make the higher vibrationalstates accessible.
5.3.5 Krotov scheme
For the Morse oscillator, the field found by the Krotov algorithm in optnctrl (figure 5.29)was as irregular as the ones produced with the eig basis set. It yielded an infidelity of 0.009after ten iterations, which took a total computation time of 25 minutes. This result could beimproved to 0.005 by performing another five iterations, and further optimization with evenlonger calculations is probably possible. However, this approach was not followed due to thismethod’s time consumption.
In the case of LiCs, the algorithm failed to converge altogether. Instead it broke off aftereight iterations with increasingly un-physical (i.e. extremely large and discontinuous) fields. Al-though a better result might have been achieved with carefully chosen settings, this underlinesthe difficulty inherent to optimal control with LiCs.
5.4 Memory and CPU consumption
After all physical results have been presented, some final remarks concerning the computationaleffort should be made. Most calculations were performed on servers of the DESY PAL cluster,
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Figure 5.29: Krotov algorithm applied to the Morse oscillator. The large diagram shows theoptimized field after ten iterations. Inset: The infidelity reached a value of 0.009.
most of which run at 3 GHz. Although MCTDH supports parallelization, only one processorcore was used at a time.
When using the HO DVR, the propagation of the Morse system in section 4.2.3 took aboutten minutes of CPU time and one MB of memory. With the sine DVR, memory consumptionstayed roughly equal, but only about one minute of CPU time was required. This highlightshow important the choice of the DVR is for the accuracy as well as the efficiency of a simula-tion. It should be mentioned that MCTDH still allocates much more memory at initiation. Thisamounted to about 250 MB in the cases considered here, but can become a serious issue whenhandling multi-dimensional systems. For example, the sample input file for benzene (modelledin ten dimensions) results in a memory allocation of more than 2 GB, which is already to muchfor a default job on DESY’s BIRD cluster.
The more important question is how CRAB fares in comparison to the Krotov scheme. Thelatter usually gives excellent results even with a low number of iterations because it calculates thefield E(t) while propagating the wavefunction, using the value of Ψ(t) for all times. In contrast,CRAB only considers the final target state population and discards all wavefunction informationfor intermediate times. In combination with the finite-dimensional search space, optimization isusually not quite as good as with the Krotov method. However, CRAB requires only very littleresources, with virtually all computational effort lying in the repeated MCTDH calls. With agood DVR and for low excited states (as was the case with the Morse oscillator in this thesis),such a propagation can be faster than five seconds.
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An optimization run with CRAB (with a fixed basis or with frequency optimization) usuallytook ten to fifteen minutes (wall time) for a limit of 200 function calls. MCTDH again occupiedabout one MB of memory. The disk space occupied by the output files depends on the parametertout telling MCTDH how often to update the output files. Since CRAB only cares about thefinal state, the output files can be kept very small and less than 10 MB of storage are occupiedfor an optimization run. Most of this is due to the electric field, which is stored as plain text data.There is no significant difference between the ten-parameter control with fixed basis functionsand the nine-parameter control which optimized the frequencies. The self-designed method forfinding occupation numbers (see section 4.2.3) can be a bottleneck, as it converts part of the psifile into plain text. In the examples presented here, this file was usually smaller than 1 MB, butit can quickly become prohibitively large for higher dimensions or small values of tout. Thisfault cannot be attributed to CRAB, but to the deliberate creation of a quick-and-dirty tool. Forfuture applications, a more efficient application for finding the target state occupation can (andwill) be written.
Since the Krotov scheme uses information aboutΨ(t), the wavefunction must be written to theoutput in very small time steps, typically tout is 0.01 fs or less. Since optcntrl also requiresthe wavefunction to be calculated in double precision, very large psi files are generated. Theoptimization of the Morse pulse filled over 700 MB in about half an hour. The LiCs optimizationhad a shorter control time, so only 130 MB were written. The calculation time was comparable,probably due to the numerical difficulty of the LiCs system. Although optcntrl is slower andwrites large files, its memory requirements are modest. Little more than 1 MB was used.
In summary, CRAB had a small advantage over the Krotov scheme in terms of speed (at leastfor the systems considered here) and was massively more efficient when it came to using diskspace (which is very likely generally true).
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6 Conclusion
This thesis has tested the applicability of the multi-configurational time-dependent Hartree method(MCTDH) to state selective control in diatomics, in particular in combination with the choppedrandom basis (CRAB) optimization scheme.
The motivation behind this work was to probe different tools for optimal control of molecularsystems. Powerful computational methods for the theoretical treatment of molecules can assistthe experimental path to laser selective chemistry.
MCTDH is a versatile scheme for propagating multi-dimensional wavepackets while includingall important correlation effects [3]. The implementation used in this thesis was version 8.4.9 ofthe Heidelberg MCTDH package [25]. As all systems treated in this thesis are one-dimensional,the MCTDH algorithm was not actually applied. Instead, the MCTDH package was used tosimply solve the one-dimensional Schrödinger equation. The decision to use this software wasbased on future applications to molecules with many degrees of freedom.
The MCTDH software was tested with a harmonic oscillator, which has the advantage of beinganalytically solvable. The wavefunction was propagated from different initial states, relaxed tofind the ground state, and excited into a coherent state. In all these cases, MCTDH reproducedthe expected results reliably.
The second step was selective excitation of a Morse oscillator which had been studied in theliterature before [15]. Starting from the ground state, the vibrational state |15〉 was excited withtrain of monochromatic light pulses. Two slightly different pathways were used to reach the finalstate, which was finally achieved in agreement with the literature. However, the desired accuracywas only reached after changing the discrete variable representation (DVR) from a harmonicoscillator basis to sine basis functions and after including a complex absorbing potential (CAP).A sine DVR turned out to be more apt for representing high vibrational states, and the CAPprevents errors arising from the wavefunction hitting the edge of the grid.
Next, optimal control was attempted with CRAB. CRAB is a general purpose optimizationalgorithm which has been applied to various examples of quantum control. The electric fieldapplied to the molecule is expanded in a truncated Fourier basis with randomized frequencies,converting the optimal control problem into a minimization of a multi-variate function.
An implementation of CRAB was written in Python, along with a script for state populationanalysis. The MCTDH package was used to propagate the system under the influence of thefield. The analysis script returned the target state population to the CRAB program, which thencomputed the field parameters for the next iteration.
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The systems under examination were the Morse oscillator above and the LiCs diatomic, whichhas a comparable potential energy function [24].
The algorithm was modified to use different frequency basis sets, including the resonant fre-quencies and the harmonic frequencies arising from the control time. These frequencies wereused both with their exact values as well as after multiplication with a random factor. In anotheroptimization run, the program was modified to also optimize the frequencies rather than keepingthem fixed.
Since CRAB makes no statement about the minimization algorithm to use, three differentexternal tools were tested. These are two implementations of the Nelder-Mead simplex (NMS)method from the scipy.optimize and the NLopt libraries [14, 13]. The latter library alsocontains the Subplex algorithm, a refinement of NMS. The success of any combination wasmeasured in terms target state occupation.
Target populations higher than 99% could be achieved for the Morse oscillator with |5〉 a tar-get state. It turned out that harmonic frequencies performed better and more reliably than theresonant frequencies in almost all cases. Randomization did not significantly improve the resultfor the Morse oscillator, although the randomized harmonic frequencies did not perform muchworse, either. The best results were found with optimized frequencies, however no randomiza-tion took place here.
An surprising discrepancy was found between the two NMS variations, as the scipy imple-mentation performed much worse than the NLopt version. The latter routinely gave good results,roughly on par with Subplex. Apparently the choice of the minimization algorithm is just as cru-cial for convergence as the correct frequency basis. On average Subplex did not perform muchbetter than NMS, although is might prove superior in a higher-dimensional search space.
LiCs proved very difficult to control, as even the transition from ground to first excited state isnot possible without populating higher states. This is due to the nearly equidistant spacing of thelowest eigenenergies. The calculations were further hindered by the fact that the dipole momentis not known for small inter-nuclear distances, so the grid had to be chosen a bit too short foraccurate propagations. These difficulties limited the arget population to just over about 40%with the tested algorithms. CRAB, using randomized harmonic frequencies, showed a slightadvantage over the frequency optimization method.
The very popular and well-probed Krotov method for optimal control was used for reference.The MCTDH package contains an implementation of this scheme, which can be considered stateof the art for optimal control. For the Morse oscillator, it showed a performance comparable tothe best basis-algorithm combinations with just a slightly (about 50%) larger time consumption.However, the resulting pulse showed a rather irregular shape with high field gradients, whichmight be undesirable for experimental reproductions. From this aspect, some fields optimizedwith CRAB were smoother and possibly easer to implement in practice. The Krotov methoddiverged when trying to optimize the LiCs pulse, which is attributed to the unpleasant propertiesof the system, not to the algorithm.
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When contrasting CRAB with the Krotov scheme, the former fared surprisingly well despiteits apparent simplicity. As few as nine to ten optimization parameters were enough to find apulse which was at least as good as the result of the more sophisticated Krotov method. Atthe same time, CRAB was faster and required hardly any disk space while the Krotov schemefilled hundreds of megabytes. For the problems considered in this thesis, CRAB is by no meansinferior to the Krotov scheme. For best results, it is usually advisable not to randomize thefrequency basis, but to let the algorithm search for the optimal frequencies.
Further research should follow various directions. Firstly, the restriction to one dimension is aserious limitation for experimental applications. If one aims to implement optimal control in thelaboratory, the extension to higher-dimensional systems will be necessary. This also applies toconsidering other potential energy surfaces, which are effectively treated as additional degreesof freedom by MCTDH. The algorithm has even been applied to conical intersections, so thereis a vast field of realistic molecular dynamics at grasp.
While MCTDH is – by design – very suitable for treating high-dimensional systems, this isnot yet entirely clear for CRAB. It has shown to work well in this case, but it still has to showwhether the simple truncated-basis approach leads to satisfactory fields for controlling morecomplex molecules. The results of this thesis encourage such an analysis.
Given the fact that different optimization algorithms – and even implementations of the samealgorithm – can exhibit huge differences in performance, a more systematic study of such algo-rithms and their applicability to typical optimal control problems seems reasonable.
The numerical treatment of the LiCs system was apparently inadequate. If a an accuratedescription of this molecule is desired, a thorough search for the best DVR is recommended, aswell as a workaround for the dipole function at low distances.
As Dr. Peter Schmelcher pointed out, a molecule at finite temperature is not in its pure groundstate, but rather in an incoherent superposition of eigenstates given by the Boltzmann distribu-tion. For real life applications it is important to not just control pure states, but also to excitesystems starting from such mixed states. For this purpose the wavefunction formalism is nolonger applicable, but instead a density operator ρ =
∑pi|Ψi〉〈Ψi| must be propagated. Im-
plementing this was unfortunately beyond the scope of this thesis. Yet, such propagations havealready been performed with the so-called ρMCTDH algorithm [?]. Hence, further studies inthis direction seem rewarding, too.
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Acknowledgements
I would never have written this thesis without many kind and helpful people whose efforts I wantto acknowledge here.
First, I would like to thank my advisor Prof. Dr. R.J. Dwayne Miller (MPSD) for giving methe opportunity to work in his group and for offering me the possibilities to pursue my academiccareer. Next, I am very thankful to my co-advisor Prof. Dr. Peter Schmelcher (ILP) for a lot ofsupport and helpful input over the past months. He suggested to use CRAB, supplied the LiCsdata and organized the contact with other researchers who would help me in my work.
Thanks to Prof. Dr. Hans-Dieter Meyer from Heidelberg for supplying the MCTDH software.I thank his post-doc Dr. Markus Schröder, who developed the optcntrl component of MCTDHand helped me when I couldn’t make it run. Thanks to Dr. Antonio Negretti (ILP), who gaveme valuable advice on implementing CRAB. Thanks to Dr. Taisuke Hasegawa and Dr. ArendDijkstra from the Miller group for looking after me and being generally helpful.
I thank my office mates Ara, Haider and Steffi for being a good company and creating anawesome and motivating environment to work in.
On a personal note, I have to thank my family, especially my parents for their amazing andongoing support in all aspects. I can’t possibly thank my girlfriend Sylvia enough, who supportedand motivated me and generally took great care of me, in particular during the final weeks ofthis thesis. Thanks to all my dear friends, who made my life enjoyable over the past few years.
To all of you, I thank you very much.
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Eigenstandigkeitserklarung
Hiermit bestätige ich, dass die vorliegende Arbeit von mir selbständig verfasst wurde und ich kei-ne anderen als die angegebenen Hilfsmittel – insbesondere keine im Quellenverzeichnis nichtbenannten Internetquellen – benutzt habe und die Arbeit von mir vorher nicht einem anderenPrüfungsverfahren eingereicht wurde. Die eingereichte schriftliche Fassung entspricht der aufdem elektronischen Speichermedium. Ich bin damit einverstanden, dass die Masterarbeit veröf-fentlicht wird.1
Hamburg, den 30. Juli 2014, Christian Ziemann
1English translation: I hereby declare that I have written this thesis by myself without using any undocumentedaids, in particular no internet sources other than the ones listed in the bibliography. I have not handed in thisthesis for any other examination process. This printed document is identical to the attached electronic version.I agree to publication of this thesis.
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